Interferometry systems and methods of using interferometry systems

ABSTRACT

In general, in one aspect, the invention features methods that include interferometrically monitoring a distance between an interferometry assembly and a measurement object along each of three different measurement axes while moving the measurement object relative to the interferometry assembly, determining values of a parameter for different positions of the measurement object from the monitored distances, wherein for a given position the parameter is based on the distances of the measurement object along each of the three different measurement axes at the given position, and deriving information about a surface figure profile of the measurement object from a frequency transform of at least the parameter values.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 USC §119(e)(1) to ProvisionalPatent Application No. 60/564,448, entitled “MULTI-AXIS INTERFEROMETERAND DATA PROCESSING FOR MIRROR MAPPING,” filed on Apr. 22, 2004 and toProvisional Patent Application No. 60/644,898, entitled “MULTI-AXISINTERFEROMETER AND DATA PROCESSING FOR MIRROR MAPPING” filed on Jan. 19,2005. The entire contents of Provisional Patent Application Nos.60/564,448 and 60/644,898 are hereby incorporated by reference.

BACKGROUND

This invention relates to interferometry systems and methods of usinginterferometry systems.

Displacement measuring interferometers monitor changes in the positionof a measurement object relative to a reference object based on anoptical interference signal. The interferometer generates the opticalinterference signal by overlapping and interfering a measurement beamreflected from the measurement object with a reference beam reflectedfrom the reference object.

In many applications, the measurement and reference beams haveorthogonal polarizations and different frequencies. The differentfrequencies can be produced, for example, by laser Zeeman splitting, byacousto-optical modulation, or internal to the laser using birefringentelements or the like. The orthogonal polarizations allow a polarizingbeam-splitter to direct the measurement and reference beams to themeasurement and reference objects, respectively, and combine thereflected measurement and reference beams to form overlapping exitmeasurement and reference beams. The overlapping exit beams form anoutput beam that subsequently passes through a polarizer.

The polarizer mixes polarizations of the exit measurement and referencebeams to form a mixed beam. Components of the exit measurement andreference beams in the mixed beam interfere with one another so that theintensity of the mixed beam varies with the relative phase of the exitmeasurement and reference beams.

A detector measures the time-dependent intensity of the mixed beam andgenerates an electrical interference signal proportional to thatintensity. Because the measurement and reference beams have differentfrequencies, the electrical interference signal includes a “heterodyne”signal having a beat frequency equal to the difference between thefrequencies of the exit measurement and reference beams. If the lengthsof the measurement and reference paths are changing relative to oneanother, e.g., by translating a stage that includes the measurementobject, the measured beat frequency includes a Doppler shift equal to2vnp/λ, where v is the relative speed of the measurement and referenceobjects, λ is the wavelength of the measurement and reference beams, nis the refractive index of the medium through which the light beamstravel, e.g., air or vacuum, and p is the number of passes to thereference and measurement objects. Changes in the phase of the measuredinterference signal correspond to changes in the relative position ofthe measurement object, e.g., a change in phase of 2π correspondssubstantially to a distance change d of λ/(2np). Distance 2d is around-trip distance change or the change in distance to and from a stagethat includes the measurement object. In other words, the phase Φ,ideally, is directly proportional to d, and can be expressed as

$\begin{matrix}{{\Phi = {20p\; k\; d}},{{{where}\mspace{14mu} k} = {\frac{2\pi\; n}{\lambda}.}}} & (1)\end{matrix}$

Unfortunately, the observable interference phase, {tilde over (Φ)}, isnot always identically equal to phase Φ. Many interferometers include,for example, non-linearities such as “cyclic errors.” Cyclic errors canbe expressed as contributions to the observable phase and/or theintensity of the measured interference signal and have a sinusoidaldependence on the change in for example optical path length 2pnd. Inparticular, a first order cyclic error in phase has for the example asinusoidal dependence on (4πpnd)/λand a second order cyclic error inphase has for the example a sinusoidal dependence on 2(4πpnd)/λ. Higherorder cyclic errors can also be present as well as sub-harmonic cyclicerrors and cyclic errors that have a sinusoidal dependence of otherphase parameters of an interferometer system comprising detectors andsignal processing electronics. Different techniques for quantifying suchcyclic errors are described in commonly owned U.S. Pat. No. 6,137,574,U.S. Pat. No. 6,252,688, and U.S. Pat. No. 6,246,481 by Henry A. Hill.

In addition to cyclic errors, there are other sources of deviations inthe observable interference phase from Φ, such as, for example,non-cyclic non-linearities or non-cyclic errors. One example of a sourceof a non-cyclic error is the diffraction of optical beams in themeasurement paths of an interferometer. Non-cyclic error due todiffraction has been determined for example by analysis of the behaviorof a system such as found in the work of J.-P. Monchalin, M. J. Kelly,J. E. Thomas, N. A. Kumit, A. Szöke, F. Zemike, P. H. Lee, and A. Javan,“Accurate Laser Wavelength Measurement With A Precision Two-BeamScanning Michelson Interferometer,” Applied Optics, 20(5), 736–757,1981.

A second source of non-cyclic errors is the effect of “beam shearing” ofoptical beams across interferometer elements and the lateral shearing ofreference and measurement beams one with respect to the other. Beamshears can be caused, for example, by a change in direction ofpropagation of the input beam to an interferometer or a change inorientation of the object mirror in a double pass plane mirrorinterferometer such as a differential plane mirror interferometer (DPMI)or a high stability plane mirror interferometer (HSPMI).

Inhomogeneities in the interferometer optics may cause wavefront errorsin the reference and measurement beams. When the reference andmeasurement beams propagate collinearly with one another through suchinhomogeneities, the resulting wavefront errors are identical and theircontributions to the interferometric signal cancel each other out. Moretypically, however, the reference and measurement beam components of theoutput beam are laterally displaced from one another, i.e., they have arelative beam shear. Such beam shear causes the wavefront errors tocontribute an error to the interferometric signal derived from theoutput beam.

Moreover, in many interferometry systems beam shear changes as theposition or angular orientation of the measurement object changes. Forexample, a change in relative beam shear can be introduced by a changein the angular orientation of a plane mirror measurement object.Accordingly, a change in the angular orientation of the measurementobject produces a corresponding error in the interferometric signal.

The effect of the beam shear and wavefront errors will depend uponprocedures used to mix components of the output beam with respect tocomponent polarization states and to detect the mixed output beam togenerate an electrical interference signal. The mixed output beam mayfor example be detected by a detector without any focusing of the mixedbeam onto the detector, by detecting the mixed output beam as a beamfocused onto a detector, or by launching the mixed output beam into asingle mode or multi-mode optical fiber and detecting a portion of themixed output beam that is transmitted by the optical fiber. The effectof the beam shear and wavefront errors will also depend on properties ofa beam stop should a beam stop be used in the procedure to detect themixed output beam. Generally, the errors in the interferometric signalare compounded when an optical fiber is used to transmit the mixedoutput beam to the detector.

Amplitude variability of the measured interference signal can be the netresult of a number of mechanisms. One mechanism is a relative beam shearof the reference and measurement components of the output beam that isfor example a consequence of a change in orientation of the measurementobject.

In dispersion measuring applications, optical path length measurementsare made at multiple wavelengths, e.g., 532 nm and 1064 nm, and are usedto measure dispersion of a gas in the measurement path of the distancemeasuring interferometer. The dispersion measurement can be used inconverting the optical path length measured by a distance measuringinterferometer into a physical length. Such a conversion can beimportant since changes in the measured optical path length can becaused by gas turbulence and/or by a change in the average density ofthe gas in the measurement arm even though the physical distance to themeasurement object is unchanged.

The interferometers described above are often components of metrologysystems in scanners and steppers used in lithography to produceintegrated circuits on semiconductor wafers. Such lithography systemstypically include a translatable stage to support and fix the wafer,focusing optics used to direct a radiation beam onto the wafer, ascanner or stepper system for translating the stage relative to theexposure beam, and one or more interferometers. Each interferometerdirects a measurement beam to, and receives a reflected measurement beamfrom, e.g., a plane mirror attached to the stage. Each interferometerinterferes its reflected measurement beams with a correspondingreference beam, and collectively the interferometers accurately measurechanges in the position of the stage relative to the radiation beam. Theinterferometers enable the lithography system to precisely control whichregions of the wafer are exposed to the radiation beam.

In many lithography systems and other applications, the measurementobject includes one or more plane mirrors to reflect the measurementbeam from each interferometer. Small changes in the angular orientationof the measurement object, e.g., corresponding to changes in thepitching and/or yaw of a stage, can alter the direction of eachmeasurement beam reflected from the plane mirrors. If leftuncompensated, the altered measurement beams reduce the overlap of theexit measurement and reference beams in each correspondinginterferometer. Furthermore, these exit measurement and reference beamswill not be propagating parallel to one another nor will their wavefronts be aligned when forming the mixed beam. As a result, theinterference between the exit measurement and reference beams will varyacross the transverse profile of the mixed beam, thereby corrupting theinterference information encoded in the optical intensity measured bythe detector.

To address this problem, many conventional interferometers include aretroreflector that redirects the measurement beam back to the planemirror so that the measurement beam “double passes” the path between theinterferometer and the measurement object. The presence of theretroreflector insures that the direction of the exit measurement isinsensitive to changes in the angular orientation of the measurementobject. When implemented in a plane mirror interferometer, theconfiguration results in what is commonly referred to as ahigh-stability plane mirror interferometer (HSPMI). However, even withthe retroreflector, the lateral position of the exit measurement beamremains sensitive to changes in the angular orientation of themeasurement object. Furthermore, the path of the measurement beamthrough optics within the interferometer also remains sensitive tochanges in the angular orientation of the measurement object.

In practice, the interferometry systems are used to measure the positionof the wafer stage along multiple measurement axes. For example,defining a Cartesian coordinate system in which the wafer stage lies inthe x-y plane, measurements are typically made of the x and y positionsof the stage as well as the angular orientation of the stage withrespect to the z axis, as the wafer stage is translated along the x-yplane. Furthermore, it may be desirable to also monitor tilts of thewafer stage out of the x-y plane. For example, accurate characterizationof such tilts may be necessary to calculate Abbé offset errors in the xand y positions. Thus, depending on the desired application, there maybe up to five degrees of freedom to be measured. Moreover, in someapplications, it is desirable to also monitor the position of the stagewith respect to the z-axis, resulting in a sixth degree of freedom.

To measure each degree of freedom, an interferometer is used to monitordistance changes along a corresponding metrology axis. For example, insystems that measure the x and y positions of the stage as well as theangular orientation of the stage with respect to the x, y, and z axes,at least three spatially separated measurement beams reflect from oneside of the wafer stage and at least two spatially separated measurementbeams reflect from another side of the wafer stage. See, e.g., U.S. Pat.No. 5,801,832 entitled “METHOD OF AND DEVICE FOR REPETITIVELY IMAGING AMASK PATTERN ON A SUBSTRATE USING FIVE MEASURING AXES,” the contents ofwhich are incorporated herein by reference. Each measurement beam isrecombined with a reference beam to monitor optical path length changesalong the corresponding metrology axes. Because the differentmeasurement beams contact the wafer stage at different locations, theangular orientation of the wafer stage can then be derived fromappropriate combinations of the optical path length measurements.Accordingly, for each degree of freedom to be monitored, the systemincludes at least one measurement beam that contacts the wafer stage.Furthermore, as described above, each measurement beam may double-passthe wafer stage to prevent changes in the angular orientation of thewafer stage from corrupting the interferometric signal. The measurementbeams may be generated from physically separate interferometers or frommulti-axes interferometers that generate multiple measurement beams.

SUMMARY

In general, in one aspect, the invention features methods that includeinterferometrically monitoring a distance between an interferometryassembly and a measurement object along each of three differentmeasurement axes while moving the measurement object relative to theinterferometry assembly, determining values of a first parameter and asecond parameter for different positions of the measurement object fromthe monitored distances, wherein for a given position the firstparameter is based on the monitored distances of the measurement objectalong each of the three different measurement axes at the givenposition, and for a given position the second parameter is based on themonitored distance of the measurement object along each of two of themeasurement axes at the given position, and deriving information about asurface figure profile of the measurement object from the first andsecond parameter values.

Implementations of the methods can include one or more of the followingfeatures and/or features of other aspects of the invention.

The second parameter can also be based on the monitored distance of themeasurement object along one of the measurement axes at anotherposition. Monitoring the distance between the interferometry assemblyand the measurement object can include simultaneously measuring alocation of the measurement object along each of the measurement axesfor each of the different positions of the measurement object.Determining the value of the second parameter for each position of themeasurement object can include calculating a difference between themonitored distance along two of the measurement axes for multipledifferent positions of the measurement object.

The methods can further include determining values of a third parameterfor different positions of the measurement object, wherein for eachposition the third parameter is based on the distance of the measurementobject along each of two of the measurement axes at that position andthe distance of the measurement object along one of the measurement axesat another position. The information about the surface figure profile ofthe measurement object can be derived also from the third parametervalues. The values of the first parameter can be determined for a firstrange of positions of the measurement object, the values of the secondparameter can be determined for a second range of positions of themeasurement object, and the values of the third parameter can bedetermined for a third range of positions of the measurement object. Thefirst, second, and third ranges of positions of the measurement objectcan be different. Deriving information about the surface figure profileof the measurement object can include determining Fourier coefficientsfor a function characterizing the surface figure profile. The Fouriercoefficients can be determined from a Fourier transform (e.g., adiscrete Fourier transform) of the values of the first, second, andthird parameters corrected for boundary offsets between the first,second, and third parameter values. The first parameter can be a seconddifference parameter (SDP). The second and third parameters can beextended second difference parameters (SDP^(e)). In some embodiments,the first parameter is given by an expression including[x _(j)(y)−x _(i)(y)]−η[x _(j)(y)−x _(j)(y)],where x_(n) corresponds to the distance between the interferometryassembly and the measurement position along the n-th measurement axis atposition y with n=i, j, k, for i≠j≠k and η is a non-zero constant. η canbe selected so that the values of the expression are not sensitive tosecond order to rotations of the measurement object about an axisorthogonal to at least one of the three measurement axes. η cancorrespond to a ratio of separation distances between the measurementaxes. The second parameter can be given by an expression comprising[x _(j)(y)−x _(i)(y)]+η[x _(j)(y)−x _(i)(y−y′)],where y′ is a non-zero constant. The third parameter can be given by anexpression comprising[x _(k)(y+y″)−x _(j)(y)]+η[x _(k)(y)−x _(j)(y)],where y″ is a non-zero constant.

The methods can further include monitoring an orientation of themeasurement object with respect to the measurement axes while moving themeasurement object relative to the interferometry assembly. Determiningvalues of the second parameter can include accounting for variations inthe orientation of the measurement object.

The measurement axes can be co-planar. In some embodiments, themeasurement axes are parallel. The relative movement of the measurementobject can be in a direction non-parallel to at least one of themeasurement axes. The direction can be orthogonal to at least one of themeasurement axes.

The methods can further include interferometrically monitoring theposition of the measurement object while moving the measurement objectrelative to the interferometry assembly. Interferometrically monitoringthe position of the measurement object can include monitoring a distancebetween another interferometry assembly and a second measurement objectalong a further axis non-parallel to at least one of the three differentmeasurement axes. The further axis can be substantially orthogonal to atleast one of the three different measurement axes.

The methods can also include using the information about the surfacefigure profile of the measurement object to improve the accuracy ofmeasurements made using the interferometry assembly.

In some embodiments, the methods include using a lithography tool toexpose a substrate with radiation passing through a mask whileinterferometrically monitoring the distance between the interferometryassembly and the measurement object, wherein the position of thesubstrate or the mask relative to a reference frame is related to thedistance between the interferometry assembly and the measurement object.The measurement object can be attached to a wafer stage configured tosupport the wafer.

The measurement object can be a plane mirror measurement object. Theinterferometer assembly or the measurement object can be attached to astage and at least one of the monitored distances is used to monitor theposition of the stage relative to a frame supporting the stage.

The methods can include using the information about the surface figureprofile of the measurement object to improve the accuracy of subsequentmeasurements made using the measurement object.

In another aspect, the invention features lithography methods for use infabricating integrated circuits on a wafer, the methods includesupporting the wafer on a moveable stage, imaging spatially patternedradiation onto the wafer, adjusting the position of the stage, andmonitoring the position of the stage using the measurement object andusing the information about the surface figure profile of themeasurement object derived using methods disclosed herein to improve theaccuracy of the monitored position of the stage.

In a further aspect, the invention features lithography methods for usein the fabrication of integrated circuits including directing inputradiation through a mask to produce spatially patterned radiation,positioning the mask relative to the input radiation, monitoring theposition of the mask relative to the input radiation using themeasurement object and using the information about the surface figureprofile of the measurement object derived using methods disclosed hereinto improve the accuracy of the monitored position of the mask, andimaging the spatially patterned radiation onto a wafer.

In another aspect, the invention features lithography methods forfabricating integrated circuits on a wafer including positioning a firstcomponent of a lithography system relative to a second component of alithography system to expose the wafer to spatially patterned radiation,and monitoring the position of the first component relative to thesecond component using the measurement object and using the informationabout the surface figure profile of the measurement object derived usingmethods disclosed herein to improve the accuracy of the monitoredposition of the first component.

In yet a further aspect, the invention features methods for fabricatingintegrated circuits including one or more of the aforementionedlithography methods.

In another aspect, the invention features methods for fabricating alithography mask that include directing a write beam to a substrate topattern the substrate, positioning the substrate relative to the writebeam, and monitoring the position of the substrate relative to the writebeam using the measurement object and using the information about thesurface figure profile of the measurement object derived using methodsdisclosed herein to improve the accuracy of the monitored position ofthe substrate.

In general, in another aspect, the invention features methods thatinclude interferometrically monitoring a distance between aninterferometry assembly and a measurement object along each of threedifferent measurement axes while moving the measurement object relativeto the interferometry assembly, determining values of a parameter fordifferent positions of the measurement object from the monitoreddistances, wherein for a given position the parameter is based on thedistances of the measurement object along each of the three differentmeasurement axes at the given position, and deriving information about asurface figure profile of the measurement object from a frequencytransform of at least the parameter values.

Implementations of the methods can include one or more of the followingfeatures and/or features of other aspects of the invention.

The frequency transform can be a Fourier transform (e.g., a discreteFourier transform). The information about the surface figure profile canbe derived from a frequency transform of values of a second parameter,wherein for a given position the second parameter is based on themonitored distances of the measurement object along each of two of themeasurement axes at the given position and the monitored distance of themeasurement object along one of the measurement axes at anotherposition.

In general, in another aspect, the invention features apparatus thatinclude a interferometer assembly configured to produce three outputbeams each including interferometric information about a distancebetween the interferometer and a measurement object along a respectiveaxis, and an electronic processor configured to determine values of afirst parameter and a second parameter for different positions of themeasurement object from the interferometric information, wherein for agiven position the first parameter is based on the distances of themeasurement object along each of the respective measurement axes at thegiven position, and for a given position the second parameter is basedon the monitored distance of the measurement object along two of therespective measurement axes at the given position and the monitoreddistance of the measurement object along one of the respectivemeasurement axes at another position, the electronic processor beingfurther configured to derive information about a surface figure profileof the measurement object from the interferometric information.

Embodiments of the apparatus can include one or more of the followingfeatures and/or features of other aspects of the invention.

The measurement object can be a plane mirror measurement object. Thethree output beams can each include a component that makes one pass tothe measurement object along a common beam path. The measurement axescan be co-planar and/or parallel.

In some embodiments, the apparatus further includes a stage, wherein theinterferometer assembly or measurement object are attached to the stage.

In another aspect, the invention features lithography systems for use infabricating integrated circuits on a wafer, where the systems include astage for supporting the wafer, an illumination system for imagingspatially patterned radiation onto the wafer, a positioning system foradjusting the position of the stage relative to the imaged radiation,and apparatus disclosed herein for monitoring the position of the waferrelative to the imaged radiation.

In general, in another aspect, the invention features lithographysystems for use in fabricating integrated circuits on a wafer, where thesystems include a stage for supporting the wafer, and an illuminationsystem including a radiation source, a mask, a positioning system, alens assembly, and apparatus disclosed herein, wherein during operationthe source directs radiation through the mask to produce spatiallypatterned radiation, the positioning system adjusts the position of themask relative to the radiation from the source, the lens assembly imagesthe spatially patterned radiation onto the wafer, and the apparatusmonitors the position of the mask relative to the radiation from thesource.

In another aspect, the invention features a beam writing systems for usein fabricating a lithography mask, where the beam writing systemsinclude a source providing a write beam to pattern a substrate, a stagesupporting the substrate, a beam directing assembly for delivering thewrite beam to the substrate, a positioning system for positioning thestage and beam directing assembly relative one another, and apparatusdisclosed herein for monitoring the position of the stage relative tothe beam directing assembly.

In other aspects, the invention features methods for fabricatingintegrated circuits that include using lithography systems disclosedherein. In general, in a further aspect, the invention featuresapparatus that include an interferometer assembly configured to producethree output beams each including interferometric information about adistance between the interferometer and a measurement object along arespective axis, and an electronic processor configured to determinevalues of a parameter or different positions of the measurement objectfrom the interferometric information, wherein for a given position theparameter is based on the distances of the measurement object along eachof the respective measurement axes at the given position, the electronicprocessor being further configured to derive information about a surfacefigure profile of the measurement object from a frequency transform ofat least the parameter values. Embodiments of the apparatus can includefeatures of other aspects of the invention.

In general, in yet another aspect, the invention features methods thatinclude interferometrically monitoring a distance between aninterferometry assembly and a measurement object along each of threedifferent measurement axes while moving the measurement object relativeto the interferometry assembly, determining values of a first parameterfor different positions of the measurement object from the monitoreddistances, wherein for a given position the first parameter based on themonitored distance of the measurement object along each of two of themeasurement axes at the given position and on the monitored distance ofthe measurement object along one of the measurement axes at anotherposition, and deriving information about a surface figure profile of themeasurement object from the first parameter values. Embodiments of themethods can include features of other aspects of the invention.

Embodiments of the invention can include one or more of the followingadvantages.

Embodiments include methods for accurately determining a surface figureerror function of a measurement object using an interferometry assembly.Where the interferometry system forms part of a metrology system in alithography tool, the interferometry system can provide in situmeasurements of measurement objects used by the metrology system whichcan reduce tool downtime needed in order to calibrate the metrologysystem. For example, the surface figure error function of a stage mirrorin a lithography tool can be monitored or measured while the tool isbeing used to expose a wafer.

The invention further provides methods for accurately determining asurface figure error function of a measurement object by performing afrequency transform of a parameter related to the second derivative ofthe surface figure error function that can be measured using amulti-axis interferometer. The methods include determining parametersthat allow use of an orthogonal set of basis functions corresponding tothe portion of the measurement object being characterized, allowingfrequency transform methods to be used.

Beams and/or axes referred to as being parallel or nominally parallelmay deviate from being perfectly parallel to the extent that the effectof the deviation on a measurement is negligible (e.g., less than therequired measurement accuracy by about an order of magnitude or more) orotherwise compensated.

Beams and/or axes referred to as being coplanar or nominally coplanarmay deviate from being perfectly coplanar to the extent that the effectof the deviation on a measurement is negligible (e.g., less than therequired measurement accuracy by about an order of magnitude or more) orotherwise compensated.

Beams and/or axes referred to as being orthogonal or nominallyorthogonal may deviate from being perfectly orthogonal to the extentthat the effect of the deviation on a measurement is negligible (e.g.,less than the required measurement accuracy by about an order ofmagnitude or more) or otherwise compensated.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. A number of references areincorporated herein by reference. In case of conflict, the presentspecification will control.

Other features and advantages of the invention will be apparent from thefollowing detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 a is a perspective drawing of an interferometer system comprisingtwo three-axis/plane interferometers.

FIG. 1 b is a diagram that shows the pattern of measurement beams frominterferometer system of FIG. 1 a at a stage mirror that serves asmeasurement object for interferometers of the interferometer system.

FIG. 2 a is a diagrammatic perspective view of an interferometer system.

FIG. 2 b is a diagram showing domains for a second difference parameter(SDP) and extended SDP-related parameters on a surface of a mirror.

FIG. 3 is a plot comparing the transfer function magnitude as a functionof spatial frequency for different values of η.

FIG. 4 is a schematic diagram of an embodiment of a lithography toolthat includes an interferometer.

FIG. 5 a and FIG. 5 b are flow charts that describe steps for makingintegrated circuits.

FIG. 6 is a schematic of a beam writing system that includes aninterferometry system.

Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

In addition to the sources of interferometer measurement errorsdiscussed previously, another source of measurement error in certaindisplacement measurement interferometry systems are variations in thesurface of a plane mirror measurement object. These variations cause themirror surface to deviate from being perfectly planar.

Errors in interferometry systems that use plane mirror measurementobjects can be reduced where a surface figure of the mirror is known.Knowledge of the surface figures allows the system to compensate for thevariations in the surface figure from an ideal mirror. However, thesurface figure of plane mirrors can vary with time, so the accuracy ofthe interferometry system measurements can degrade over the system'slifetime. Accordingly, the surface figures of the mirrors used in ametrology system should be re-measured periodically to maintain systemaccuracy.

Accurate knowledge of the surface figure of a plane mirror measurementobject can be particularly beneficial where metrology systems are usedin applications with high accuracy requirements. An example of such anapplication is in lithography tools where metrology systems are used tomonitor the position of stages that support a wafer or a mask within thetool. In some embodiments, ex situ measurement methods can be used todetermine information about the surface figure of a mirror. In thesemethods, the mirror is removed from the lithography tool and measuredusing another piece of apparatus. However, the metrology system cannotbe used until the mirror is replaced, so the tool is generallyunproductive during such maintenance. Furthermore, the surface figure ofa mirror can change when the tool is reinserted into the tool (e.g., asa result of stresses associated with the mechanical attachment of themirror to the tool), introducing unaccounted sources of error into themetrology system.

In situ measurement methods can mitigate these errors because thesurface figure is measured while it is attached to the tool, after themirror has adapted to stresses associated with its attachment to thetool. Moreover, in situ methods can improve tool throughput by reducingthe amount of time a tool is offline for servicing.

In this application, interferometry systems and methods are disclosed inwhich information about a surface figure of a mirror can be obtained ina lithography tool during the operation of the tool or while the tool isoffline. More generally, the systems and methods are not limited to usein lithography tools, and can be used in other applications as well(e.g., in beam writing systems).

In embodiments, procedures for determining information about a surfacefigure of a stage mirror includes measuring values of a seconddifference parameter (SDP) for the mirror. The SDP of a mirror can beexpressed as a series of orthogonal basis functions (e.g., a Fourierseries), where the coefficients (e.g., Fourier coefficients) of theseries are related by a transfer function to corresponding coefficientsof a series representation of a mirror surface figure error function, ξ.

However, the SDP function is not necessarily mathematically invertibleto obtain a complete set of orthogonal basis functions used in a seriesrepresentation. Accordingly, additional functions are defined that,together with the SDP, allow use of a set of orthogonal basis functionswhich permit inversion of the SDP to a conjugate (e.g., spatialfrequency) domain. These additional functions are referred to asextended SDP-related parameters (SDP^(e)s).

Before discussing the mathematical details of inverting a SDP to obtainξ, a description of an apparatus is presented that can be used toacquire data from which SDP and SDP^(e) values can be calculated.Measurements of SDP values and SDP^(e) values can be made using amulti-axis/plane interferometer. FIG. 1 a shows an embodiment of amulti-axis plane mirror interferometer 100, which directs multiplemeasurement beams to each contact a measurement object 120 (e.g., aplane mirror measurement object) twice. Interferometer 100 producesmultiple output beams 181–183 and 191–193 each including interferometricinformation about changes in distance between the interferometry systemand the measurement object along a corresponding measurement axis.

Interferometer 100 has the property that the output beams each includesa measurement component that makes one pass to the measurement objectalong one of two common measurement beam paths before being directedalong separate measurement beam paths for the second pass to themeasurement object. Similar interferometers are disclosed in commonlyowned U.S. patent application Ser. No. 10/351,707 by Henry, A. Hillfiled Jan. 27, 2003 and entitled “MULTIPLE DEGREE OF FREEDOMINTERFEROMETER,” the contents of which are incorporated herein byreference.

Interferometer 100 includes a non-polarizing beam splitter 110, whichsplits a primary input beam 111 into two secondary input beams 112A and112B. Interferometer 100 also includes a polarizing beam splitter 115,which splits secondary input beams 112A and 112B into primarymeasurement beams 113A and 113B, and primary reference beams 114A and114B, respectively. Interferometer 100 directs primary measurement beams113A and 113B along paths that contact measurement object 120 atdifferent locations in a vertical direction. Similarly, primaryreference beams 114A and 114B are directed along reference beam pathsthat contact a reference mirror 130 at different locations.Interferometer 100 also includes quarter wave plates 122 and 124.Quarter wave plate 122 is located between polarizing beam splitter 115and measurement object 120, while quarter wave plate 124 is locatedbetween polarizing beam splitter 115 and the reference mirror. Thequarter wave plates rotate by 90° the polarization state of doublepassed beams directed between the polarizing beam splitter and themeasurement object or reference mirror. Accordingly, the polarizing beamsplitter transmits an incoming beam that would have been reflected inits out-going polarization state.

The following description pertains to primary measurement beam 113A andprimary reference beam 114A. Interferometer 100 directs measurement beam113B and reference beam 114B along analogous paths. Polarizing beamsplitter (PBS) 115 transmits reflected primary measurement beam 113B,which is reflected back towards PBS 115 by a retroreflector 140 (asimilar retroreflector 141 reflects primary measurement beam 113B). Acompound optical component 150 including non-polarizing beam splitters151 and 152 and reflector 153 split primary measurement beam 113A intothree secondary measurement beams 161, 162, and 163. PBS 115 transmitsthe three secondary measurement beams, which propagate along paths thatcontact measurement object 120 at three different positions in ahorizontal plane shared by primary measurement beam 113A. PBS 115 thendirects the three secondary measurement beams reflected from measurementobject 120 along output paths.

PBS 115 reflects primary reference beam 114A towards retroreflector 140.As for the primary measurement beam, optical component 150 splitsprimary reference beam 114A reflected by retroreflector 140 into threesecondary reference beams 171, 172, and 173. PBS 115 reflects secondaryreference beams 171, 172, and 173 towards reference mirror 130 alongpaths at three different positions in a plane shared by primaryreference beam 114A. PBS 115 transmits secondary reference beams 171,172, and 173 reflected from reference object 130 along output paths sothat they overlap with measurement beams 161, 162, and 163 to formoutput beams 181, 182, and 183, respectively. The phase of the outputbeams carries information about the position of the measurement objectalong three measurement axes defined by the primary measurement beam'spath and the secondary measurement beams' paths.

Interferometer 100 also includes a window 160 located between quarterwave plate 122 and measurement object 120.

The pattern of measurement beams incident on a plane mirror measurementobject is shown in FIG. 1 b. The angle of incidence of measurement beamsat the mirror surface is nominally zero when the measurement axes areparallel to the x-axis of a coordinate system. The locations of themeasurement axes of the top multiple-axis/plane interferometercorresponding to x₁, x₂, and x₃ are shown in FIG. 1 b. The spacingsbetween measurement axes corresponding to x₁ and x₂ and to x₁ and x₃ areb₂ and b₃, respectively. In general, b₂ and b₃ can vary as desired. b₂can be the same as or different from (b₃−b₂). In some embodiments, theaxis spacing can be relatively narrow (e.g., about 10 cm or less, about5 cm or less, about 3 cm or less, about 2 cm or less). For example,where the resolution of a measurement depends on the spacing the axes,having relatively narrow spacing between at least two of the measurementaxes can provide increased sensitivity to higher frequencies in ameasurement.

Also shown in FIG. 1 b is the location corresponding to the primarysingle pass measurement beam x′₀ and the locations corresponding to thesecond pass measurement beams x′₁, x′₂, and x′₃. The relationshipbetween a linear displacement measurement corresponding to a double passto the stage mirror and the linear displacement measurementscorresponding to a single pass to the stage mirror is given by

$\begin{matrix}{{x_{j} = {\frac{1}{2}\left( {x_{j}^{\prime} + x_{0}^{\prime}} \right)}},\mspace{31mu}{j = 1},2,{{and}\mspace{14mu} 3.}} & (2)\end{matrix}$

The difference between two linear displacements x_(i) and x_(j), i≠j,referred to as a first difference parameter (FDP) is independent of x′₀,i.e.,

$\begin{matrix}{{{x_{i} - x_{j}} = {\frac{1}{2}\left( {x_{i}^{\prime} - x_{j}^{\prime}} \right)}},\mspace{31mu} i,{j = 1},2,{{and}\mspace{14mu} 3},\mspace{14mu}{i \neq {j.}}} & (3)\end{matrix}$

Reference is made to FIG. 2 which is a diagrammatic perspective view ofan interferometry system 15 that employs a pair of orthogonally arrangedinterferometers or interferometer subsystems by which the shape ofon-stage mounted stage mirrors may be characterized in situ along one ormore datum lines. As shown in FIG. 2, system 15 includes a stage 16 thatcan form part of a microlithography tool. Affixed to stage 16 is a planestage mirror 50 having a y-z reflective surface 51 elongated in they-direction.

Also, affixed to stage 16 is another plane stage mirror 60 having an x-zreflective surface 61 elongated in the x-direction. Mirrors 50 and 60are mounted on stage 16 so that their reflective surfaces, 51 and 61,respectively, are nominally orthogonal to one another. Stage 16 isotherwise mounted for translations nominally in the x-y plane but mayexperience small angular rotations about the x, y, and z axes due to,e.g., bearing and drive mechanism tolerances. In normal operation,system 15 is adapted to be operated for scanning in the y-direction forset values of x.

Fixedly mounted off-stage (e.g., to a frame of the lithography tool) isan interferometer (or interferometer subsystem) that is generallyindicated at 10. Interferometer 10 is configured to monitor the positionof reflecting surface 51 along three measurement axes (e.g., x₁, x₂, andx₃) in each of two planes parallel to the x-axis, providing a measure ofthe position of stage 16 in the x-direction and the angular rotations ofstage 16 about the y-and z-axes as stage 16 translates in they-direction. Interferometer 10 includes two three-axis/planeinterferometers such as interferometer 100 shown in FIG. 1 a andarranged so that interferometric beams travel to and from mirror 50generally along optical paths designated generally as path 12.

Also fixedly mounted off-stage (e.g., to the frame of the lithographytool) is an interferometer (or interferometer subsystem) that isgenerally indicated at 20. Interferometer 20 can be used to monitor theposition of reflecting surface 61 along three measurement axes (e.g.,y₁, y₂, and y₃) in each of two planes parallel to the y-axis, providinga measure of the position of stage 16 in the y-direction and the angularrotations of stage 16 about the x-and z-axes as stage 16 translates inthe x-direction. Interferometer 20 includes two three-axis/planeinterferometers such as interferometer 100 shown in FIG. 1 a andarranged so that interferometric beams travel to and from mirror 60generally along an optical path designated as 22.

In general, simultaneously sampled values for x₁, x₂, and x₃ for x-axisstage mirror 50 are acquired as a function of position of the mirror inthe y-direction with the corresponding x-axis location and the stagemirror orientation nominally held at fixed values. In addition, valuesfor y₁, y₂, and y₃ for y-axis stage mirror 60 are measured as a functionof the position in the x-direction of the y-axis stage mirror with thecorresponding y-axis location and stage orientation nominally held atfixed values. The orientation of the stage can be determined initiallyfrom information obtained using one or both of the three-axis/planeinterferometers.

The SDP and the SDP^(e) values for the x-axis stage mirror arecalculated as a function of position of the x-axis stage mirror in they-direction with the corresponding x-axis location and the stage mirrororientation nominally held at fixed values from the data acquired asdiscussed previously. Also the SDP and the SDP^(e) values for the y-axisstage mirror are measured as a function of position in the x-directionof the y-axis stage mirror with the corresponding y-axis location andstage orientation nominally held at fixed values. Increased sensitivityto high spatial frequency components of the surface figure of a stagemirror can be obtained by measuring the respective SDP and the SDP^(e)values with the stage oriented at large yaw angles and/or largemeasurement path lengths to the stage mirror, i.e., for largemeasurement beam shears at the respective measuring three-axis/planeinterferometer.

The measurements of the respective SDP values for the x-axis and y-axisstage mirrors do not require monitoring of changes in stage orientationduring the respective scanning of the stage mirrors other than tomaintain the stage at a fixed nominal value since SDP is independent ofstage mirror orientation except for third order effects. However,accurate monitoring of changes in stage orientation during scanning of astage mirror may be required when a three-axis/plane interferometer isused to measure SDP^(e)s since these parameters may be sensitive tochanges in stage orientation.

The surface figure error function for an x-axis stage mirror will, ingeneral, be a, function of both y and z and the surface figure errorfunction for the y-axis stage mirror will, in general, be a function ofboth x and z. The z dependence of surface figure error functions issuppressed in the remaining description of the data processingalgorithms except where explicitly noted. The generalization to coverthe z dependent properties can be addressed by using twothree-axis/plane interferometer system in conjunction with a procedureto measure the angle between the x-axis and y-axis stage mirrors in twodifferent parallel planes displaced along the z-direction correspondingto the two planes defined by the two three-axis/plane interferometersystem.

The following description is given in terms of procedures used withrespect to the x-axis stage mirror since the corresponding algorithmsused with respect to the y-axis stage mirror can be easily obtained by atransformation of indices. Once the system has acquired values forx₁(y), x₂(y), and x₃(y) for one or more locations of the stage withrespect to the x-axis, values for SDP and on or mote SDP^(e)s arecalculated from which ξ(y) can be determined.

Turning now to mathematical expressions for the SDP and SDP, SDP^(e) canbe defined for a three-axis/plane interferometer such that it is notsensitive to either a displacement of a respective mirror or to therotation of the mirror except through a third and/or higher order effectinvolving the angle of stage rotation in a plane defined by thethree-axis/plane interferometer and departures of the stage mirrorsurface from a plane.

Different combinations of displacement measurements x₁, x₂, and x₃ maybe used in the definition of a SDP. One definition of a SDP for anx-axis stage mirror is for example

$\begin{matrix}{{{{SDP}(y)} \equiv {\left( {x_{2} - x_{1}} \right) - {\frac{b_{2}}{b_{3} - b_{2}}\left( {x_{3} - x_{2}} \right)}}}{or}} & (4) \\{{{{SDP}(y)} = {\left( {x_{2} - x_{1}} \right) - {\eta\left( {x_{3} - x_{2}} \right)}}}{where}} & (5) \\{\eta \equiv {\frac{b_{2}}{b_{3} - b_{2}}.}} & (6)\end{matrix}$Note that SDP can be written in terms of single pass displacements usingEquation (3), i.e.,

$\begin{matrix}{{{SDP}(y)} = {{\frac{1}{2}\left\lbrack {\left( {x_{2}^{\prime} - x_{1}^{\prime}} \right) - {\eta\left( {x_{3}^{\prime} - x_{2}^{\prime}} \right)}} \right\rbrack}.}} & (7)\end{matrix}$

Of course, corresponding equations apply for a y-axis stage mirror.

The lowest order term in a series representation of the SDP is relatedto the second spatial derivative of a functional representation of thesurface figure (e.g., corresponding to the second order term of a Taylorseries representation of a surface figure error function). Accordingly,for relatively large spatial wavelengths, the SDP is substantially equalto the second derivative of the mirror surface figure. Generally, theSDP is determined for a line intersecting a mirror in a plane determinedby the orientation of an interferometry assembly with respect to thestage mirror.

Certain properties of the three-axis/plane interferometer relevant tomeasurement of values of SDP are apparent. For example, SDP isindependent of a displacement of the mirror for which SDP is beingmeasured. Furthermore, SDP is generally independent of a rotation of themirror for which SDP is being measured except through a third order orhigher effects. SDP should be independent of properties of the primarysingle pass measurement beam path x′₀ in the three-axis/planeinterferometer. SDP should be independent of properties of theretroreflector in the three-axis/plane interferometer. Furthermore, SDPshould be independent of changes in the average temperature of thethree-axis/plane interferometer and should be independent of lineartemperature gradients in the three-axis/plane interferometer. SDP shouldbe independent of linear spatial gradients in the refractive indices ofcertain components in a three-axis/plane interferometer. SDP should beindependent of linear spatial gradients in the refractive indices and/orthickness of cements between components in a three-axis/planeinterferometer. SDP should also be independent of “prism effects”introduced in the manufacture of components of a three-axis/planeinterferometer used to measure the SDP.

SDP^(e)s are designed to exhibit spatial filtering properties of thesurface figure error function ξ that are similar to the spatialfiltering properties of the surface figure error function ξ by SDP for agiven portion of the respective stage mirror. SDP^(e)s can also bedefined that exhibit invariance properties similar to those of SDP inprocedures for determining the surface figure error function ξ. As aconsequence of the invariance properties, neither knowledge of eitherthe orientation of the stage mirrornor accurate knowledge of position ofthe stage mirror is required, e.g., accurate knowledge is not requiredof the position in the direction of the displacement measurements usedto compute the SDP^(e)s. However, changes in the orientation of thestage mirror are monitored during the acquisition of linear displacementmeasurements used to determine SDP^(e)s.

SDP^(e)s can be defined such that the properties of the transferfunctions with respect to the surface figure error function ξ is similarto the properties of the transfer function of SDP with respect to thesurface figure error function ξ. The properties of SDP^(e)s with respectto terms of linear first pass displacement measurements is developedfrom knowledge of the properties of the SDP with respect to terms oflinear first pass displacement measurements for a surface error functionξ that is periodic with a fundamental periodicity length L and fordomains in y that are exclusive of the domain in y for which thecorresponding SDP is valid. Two SDP^(e)s, denoted as SDP₁ ^(e) and SDP₂^(e), for a domain given byL=q2b ₃ , q=2,3, . . . ,  (8)

are given by the formulae

$\begin{matrix}{{{S\; D\;{P_{1}^{e}\left( {y,\vartheta_{z}} \right)}} \equiv {{\frac{1}{2}\begin{Bmatrix}{\left\lbrack {{x_{2}^{\prime}(y)} - {x_{1}^{\prime}(y)}} \right\rbrack +} \\{\eta\left\lbrack {{x_{2}^{\prime}(y)} - {x_{1}^{\prime}\left( {y - L + {2b_{3}}} \right)}} \right\rbrack}\end{Bmatrix}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {S\; D\;{P_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)}} \right\rbrack}}}},\mspace{79mu}{{\frac{1}{2} - \frac{2\left( {b_{3} - b_{2}} \right)}{L}} \leq \frac{y}{L} \leq \frac{1}{2}},} & (9) \\{{{{S\; D\;{P_{2}^{e}\left( {y,\vartheta_{z}} \right)}} \equiv {{{- \frac{1}{2}}\left\{ {\left\lbrack {{x_{3}^{\prime}\left( {y + L - {2b_{3}}} \right)} - {x_{2}^{\prime}(y)}} \right\rbrack + {\eta\left\lbrack {{x_{3}^{\prime}(y)} - {x_{2}^{\prime}(y)}} \right\rbrack}} \right\}} - \mspace{11mu}{x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {S\; D\;{P_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)}} \right\rbrack}}}},{{- \mspace{275mu}\frac{1}{2}} \leq \frac{y}{L} \leq {{- \frac{1}{2}} + {\frac{2b_{2}}{L}.}}}}\mspace{185mu}} & (10)\end{matrix}$where the last term in Equations (9) and (10) is a third order term withan origin in a second order geometric term such as described in commonlyowned U.S. patent application Ser. No. 10/347,271 entitled “COMPENSATIONFOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRRORINTERFEROMETERS” and U.S. patent application Ser. No. 10/872,304entitled “COMPENSATION FOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS INPLANE MIRROR INTERFEROMETER METROLOGY SYSTEMS,” both of which are byHenry A. Hill and both of which are incorporated herein in theirentirety by reference.

Expressing SDP₁ ^(e) and SDP₂ ^(e) in terms of pairs of single passdisplacement measurements obtained at time t_(j), j=1, 2, . . . , thefollowing equations are derived from Equations (9) and (10):

$\begin{matrix}{{{{S\; D\;{P_{1}^{e}\left( {y,\vartheta_{z}} \right)}} \equiv}\quad}{\quad{\frac{1}{2}{\quad{{{\left( {1 + \eta} \right)\left\lbrack {{x_{2}^{\prime}\left( {y,t_{1}} \right)} - {x_{1}^{\prime}\left( {y,t_{1}} \right)}} \right\rbrack} + {\eta\begin{Bmatrix}\left\lbrack {{x_{3}^{\prime}\left( {{y - {2b_{3}}},t_{2}} \right)} - {x_{1}^{\prime}\left( {{y - {2b_{3}}},t_{2}} \right)}} \right\rbrack \\{+ \left\lbrack {{x_{3}^{\prime}\left( {{y - {2 \times 2b_{3}}},t_{3}} \right)} - {x_{1}^{\prime}\left( {{y - {2 \times 2b_{3}}},t_{3}} \right)}} \right\rbrack} \\\vdots \\{+ \left\lbrack {{x_{3}^{\prime}\left( {{y - {\left( {q - 1} \right)2b_{3}}},t_{q}} \right)} - {x_{1}^{\prime}\left( {{y - {\left( {q - 1} \right)2b_{3}}},t_{q}} \right)}} \right\rbrack}\end{Bmatrix}} - \mspace{166mu}{x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {S\; D\;{P_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)}} \right\rbrack}}},{{\frac{1}{2} - \frac{2\left( {b_{3} - b_{2}} \right)}{L}} \leq \frac{y}{L} \leq \frac{1}{2}},}}}}} & (11) \\{{{{S\; D\;{P_{2}^{e}\left( {y,\vartheta_{z}} \right)}} = {{{- \frac{1}{2}}{\left( {1 + \eta} \right)\left\lbrack {{x_{3}^{\prime}\left( {y,t_{1}} \right)} - {x_{2}^{\prime}\left( {y,t_{1}} \right)}} \right\rbrack}} + {\begin{Bmatrix}{+ \left\lbrack {{x_{3}^{\prime}\left( {{y + L - {2b_{3}}},t_{2}} \right)} - {x_{1}^{\prime}\left( {{y + L - {2b_{3}}},t_{2}} \right)}} \right\rbrack} \\{+ \left\lbrack {{x_{3}^{\prime}\left( {{y + L - {2 \times 2b_{3}}},t_{3}} \right)} - {x_{1}^{\prime}\left( {{y + L - {2 \times 2b_{3}}},t_{3}} \right)}} \right\rbrack} \\\vdots \\{+ \left\lbrack {{x_{3}^{\prime}\left( {{y + L - {\left( {q - 1} \right)2b_{3}}},t_{q}} \right)} - {x_{1}^{\prime}\left( {{y + L - {\left( {q - 1} \right)2b_{3}}},t_{q}} \right)}} \right\rbrack}\end{Bmatrix}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {S\; D\;{P_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)}} \right\rbrack}}}},{{- \frac{1}{2}} \leq \frac{y}{L} \leq {{- \frac{1}{2}} + \frac{2b_{2}}{L}}},{{{where}\mspace{14mu} t_{i}} \neq t_{j}},{i \neq {j.}}}\mspace{169mu}} & (12)\end{matrix}$

FIG. 2 b illustrate the relationship of the domains for which SDP, SDP₁^(e), and SDP₂ ^(e) are defined on mirror surface 51. Domain Lcorresponds to a portion of mirror surface 51 along a scan line 201 inthe y-direction. The locations of the first pass beams on mirror surface51 are shown for the mirror at position y_(i). Note that the domains iny for SDP, SDP₁ ^(e), and SDP₂ ^(e), i.e.,

${{{- \frac{1}{2}} + \left( \frac{2b_{2}}{L} \right)} \leq \frac{y}{L} \leq {\frac{1}{2} - \left( \frac{{2b_{3}} - {2b_{2}}}{L} \right)}},{{\frac{1}{2} - \frac{\left( {{2b_{3}} - {2b_{2}}} \right)}{L}} \leq \frac{y}{L} \leq \frac{1}{2}},{{{and} - \frac{1}{2}} \leq \frac{y}{L} \leq {{- \frac{1}{2}} + \frac{2b_{2}}{L}}},$are mutually exclusive and that the combined domains in y of the threedomains cover the domain

${- \frac{1}{2}} \leq \frac{y}{L} \leq {\frac{1}{2}.}$

For a section of the x-axis stage mirror covering domain L in y, thesurface figure error function ξ(y) can be expressed by the Fourierseries

$\begin{matrix}{{{\xi(y)} = {{\sum\limits_{m = 0}^{N}{A_{m}{\cos\left( {m\; 2\pi\;\frac{y}{L}} \right)}}} + {\sum\limits_{m = 0}^{N}{B_{m}{\sin\left( {m\; 2\pi\;\frac{y}{L}} \right)}}}}},{{- \frac{1}{2}} \leq \frac{y}{L} \leq \frac{1}{2}},} & (13)\end{matrix}$

where N is an integer determined by consideration of the spatialfrequencies that are to be included in the series representation. Theterm represented by A₀ which is sometimes referred to as a “piston” typeerror is included in Equation (13) for completeness. An error of thistype is equivalent to the effect of a displacement of the stage mirrorin the direction orthogonal to the stage mirror surface and as such isnot considered an intrinsic property of the surface figure errorfunction. Using the definition of SDP given by Equation (7), thecorresponding series for SDP is next written as

$\begin{matrix}{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {\frac{1}{2}{\sum\limits_{m = 1}^{N}\;{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)} \times}}}} & (14) \\{\mspace{166mu}{\begin{Bmatrix}{A_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{2b_{2}}{L}} \right)} -} \right.} \\{\left. {{\eta cos}\left( {m\; 2\;\pi\frac{{2b_{3}} - {2b_{2}}}{L}} \right)} \right\rbrack +} \\{B_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{2b_{2}}{L}} \right)} - {{\eta sin}\left( {m\; 2\;\pi\frac{{2b_{3}} - {2b_{2}}}{L}} \right)}} \right\rbrack}\end{Bmatrix} +}} & \; \\{\mspace{160mu}{\frac{1}{2}{\sum\limits_{m = 1}^{N}\;{{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)} \times}}}} & \; \\{\mspace{155mu}{\begin{Bmatrix}{{- {A_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{2b_{2}}{L}} \right)} - {{\eta sin}\left( {m\; 2\;\pi\frac{{2b_{3}} - {2b_{2}}}{L}} \right)}} \right\rbrack}} +} \\{B_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{2b_{2}}{L}} \right)} -} \right.} \\\left. {{\eta cos}\left( {m\; 2\;\pi\frac{{2b_{3}} - {2b_{2}}}{L}} \right)} \right\rbrack\end{Bmatrix} -}} & \; \\{\mspace{169mu}{{x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}},}} & \; \\{\mspace{149mu}{{{{- \frac{1}{2}}\left( {1 - \frac{2\; b_{2}}{L}} \right)} \leq \frac{y}{L} \leq {\frac{1}{2}\left\lbrack {1 - \left( \frac{{2\; b_{3}} - {2\; b_{2}}}{L} \right)} \right\rbrack}},}} & \;\end{matrix}$

where x is a linear displacement of the stage mirror based on one ormore of the linear displacements x_(i), i=1,2, and/or 3, and θ_(z) isthe angular orientation of the stage mirror in the x-y plane. The lastterm in Equation (14) is a third order term with an origin in a secondorder geometric term such as described in commonly owned U.S. patentapplication Ser. No. 10/347,271 entitled “COMPENSATION FOR GEOMETRICEFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETERS” and U.S.patent application Ser. No. 10/872,304 entitled “COMPENSATION FORGEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRROR INTERFEROMETERMETROLOGY SYSTEMS,” both of which are by Henry A. Hill and both of whichare incorporated herein in their entirety by reference.

The presence of the third order term in Equation (14) makes it possibleto measure the high spatial frequency components of ξ(y) in the x-yplane with increased sensitivity compared to the sensitivity representedby the first order, i.e., remaining, terms in Equation (14). The thirdorder term also makes it possible to obtain information aboutintermediate and low spatial frequency components of ξ in the x-z planecomplimentary to that obtained by the first order terms in Equation(14).

Note in Equation (14) that the constant term A₀ is not present. Thisproperty is subsequently used in a procedure for elimination of effectsof the offset errors and changes in stage orientation. The loss ofinformation about the constant term A₀ is not relevant to thedetermination of the intrinsic portion of the surface figure errorfunction, since as noted above, A₀ corresponds to a displacement of thestage mirror in the direction orthogonal to the surface of the stagemirror.

Equation (14) may be rewritten in terms of η and b₃ and eliminating b₂by using Equation (6) with the result

$\begin{matrix}{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {\frac{1}{2}{\sum\limits_{m = 1}^{N}\;{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)} \times}}}} & (15) \\{\mspace{175mu}{\begin{Bmatrix}{A_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} -} \right.} \\{\left. {{\eta cos}\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)} \right\rbrack +} \\{B_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} -} \right.} \\\left. {{\eta sin}\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)} \right\rbrack\end{Bmatrix} +}} & \; \\{{\frac{1}{2}{\sum\limits_{m = 1}^{N}\;{{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)} \times}}}} & \; \\{\mspace{175mu}{\begin{Bmatrix}{- {A_{m}\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} -} \right.}} \\{\left. {{\eta sin}\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)} \right\rbrack +} \\{B_{m}\left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} -} \right.} \\\left. {{\eta cos}\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)} \right\rbrack\end{Bmatrix} -}} & \; \\{\mspace{185mu}{{x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}},}} & \; \\{\mspace{175mu}{{- {\frac{1}{2}\left\lbrack {1 - {\left( \frac{\eta}{1 + \eta} \right)\frac{2\; b_{3}}{L}}} \right\rbrack}} \leq \frac{y}{L} \leq {{\frac{1}{2}\left\lbrack {1 - {\left( \frac{1}{1 + \eta} \right)\frac{2\; b_{3}}{L}}} \right\rbrack}.}}} & \;\end{matrix}$

A contracted form of Equation (15) is obtained with the introduction ofa complex transfer function T(m) having real and imaginary amplitudes ofT_(Re) and T_(Im), respectively, as

$\begin{matrix}{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack \;{{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (16) \\{\mspace{175mu}{{x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}},}} & \; \\{\mspace{169mu}{{- {\frac{1}{2}\left\lbrack {1 - {\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack}} \leq \frac{y}{L} \leq {\frac{1}{2}\left\lbrack {1 - {\left( \frac{1}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack}}} & \; \\{where} & \; \\{A_{m}^{\prime} = {{A_{m}T_{Re}} + {B_{m}T_{Im}}}} & (17) \\{and} & \; \\{B_{m}^{\prime} = {{{- A_{m}}T_{Im}} + {B_{m}{T_{Re}.}}}} & (18) \\{with} & \; \\{{T_{Re} = \left\lbrack {\left( {1 + \eta} \right) - {\cos\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} - {{\eta cos}\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)}} \right\rbrack},} & (19) \\{and} & \; \\{T_{Im} = {\left\lbrack {{\sin\left( {m\; 2\;\pi\frac{\eta}{1 + \eta}\frac{2b_{3}}{L}} \right)} - {{\eta sin}\left( {m\; 2\;\pi\frac{1}{1 + \eta}\frac{2b_{3}}{L}} \right)}} \right\rbrack.}} & (20)\end{matrix}$

The transfer function, T(m), relates the Fourier coefficients A′_(m) andB′_(m) of the SDP to the Fourier coefficients of ξ(y), A_(m) and B_(m).As is evident from Equations (19) and (20), T(m) depends on η and b₃/Lin addition to depending on spatial frequency (as indicated bydependence on m). The dependence on spatial frequency manifests as avarying sensitivity of the transfer function to spatial frequency, withzero sensitivity occurring at certain spatial frequencies. Lack ofsensitivity at a spatial frequency means that the SDP parameter does notcontain any information of the mirror surface for that frequency.Subsequent calculation of the function ξ(y) should make note of theselow sensitivity frequencies and handle them appropriately.

As an example, the magnitude |T(m)| of T(m) is plotted in FIG. 3 as afunction of m for L=4b₃ and 2b₃=50 mm and for η=7/4, 9/5, and 11/6. Thefirst zero in |T(m)| for m≧1, corresponding to zero sensitivity of thetransfer function, occurs at m=22, 28, and 34 for η=7/4, 9/5, and 11/6,respectively. The spatial wavelength Λ₁=4.55, 3.57, and 2.94 mm form=22, 28, and 34, respectively. The value of η is selected based onconsideration of the corresponding spatial wavelength Λ₁, whether or notthe displacements x_(i) are to be not sensitivity to spatial wavelengthΛ₁ corresponding to the first zero of |T (m)|, and the diameter of themeasurement beams.

There are values of η such as η=7/4, 9/5, and 11/6 belong to one set ofη's that lead to a property of measured values of x_(i) wherein thex_(i) are obtained using a corresponding three-axis/planeinterferometer. The first set of η's can be expressed as a ratio of twointegers that can not be reduced to a ratio of smaller integers and thenumerator is an odd integer. The second set of η's are a subset of thefirst set of η's wherein the denominator of the ratio is an eveninteger. An η of the first set of η's is defined, for example, by theratio

$\begin{matrix}{{\eta = \frac{{2\; n} - 1}{n}},{n = 2},3,\;{\ldots\mspace{11mu}.}} & (21)\end{matrix}$

An η of the corresponding second set of η's is a subset of the first setof η's defined by Equation (21) for even values of η.

A property associated with this first set of η's is that the effects ofthe spatial frequencies corresponding the first zero of |T(m)| for m>0,i.e., corresponding to the spatial frequency that is not measured,cancel out in the linear displacements x₁ and x₂.

A property associated with the second set of η's is that the effects ofthe spatial frequencies corresponding the first zero of |T(m)| for m>0cancel out in each of the linear displacements x₁, x₂, and x₃.

The first zero in |T(m)| for m≧1 and for the first set of η's is givenby twice the sum of the numerator and denominator of the ratio definingan T. For the η's defined by Equation (21), the corresponding values ofm for the first zero in |T(m)| for m>0 is given by the formulam=2(3n−1).  (22)

The spatial wavelength Λ₁ corresponding to the first zero in |T(m)| form≧1 is equal to 2b₃ divided by the sum of the numerator and denominatorof the ratio defining an η. For the η's defined by Equation (21), thecorresponding values of Λ₁ are

$\begin{matrix}{\Lambda_{1} = {\frac{2\; b_{3}}{{3n} - 1}.}} & (23)\end{matrix}$

The effects of the corresponding spatial frequency cancel out in x₁ andx₂ for η's of the first set of η's because the spacing b₂ between theprimary first pass measurement beam and the second pass measurementbeams of measurement axes x₁ and x₂ are an odd harmonic of Λ₁/2, i.e.,

$\begin{matrix}{b_{2} = {\left( {{2n} - 1} \right){\frac{\Lambda_{1}}{2}.}}} & (24)\end{matrix}$

The effects of the corresponding spatial frequency cancel out in x₁, x₂,and x₃ for η's of the second set of η's because the spacing b₂ betweenthe primary first pass measurement beam and the second pass measurementbeams of measurement axes x₁ and x₂ and the spacing (2b₃−b₂) between theprimary first pass measurement beam and the second pass measurement beamof measurement axis x₃ are each an odd harmonic of Λ₁/2, i.e., spacingb₂ is given by Equation (24) and spacing (2b₃−b₂) is given by theequation

$\begin{matrix}{\left( {{2b_{3}} - b_{2}} \right) = {\left( {{4n} - 1} \right){\frac{\Lambda_{1}}{2}.}}} & (25)\end{matrix}$

There are other values of η's that belong to the first set which aregiven by the formulae

$\begin{matrix}{{\eta = \frac{{2\; n} + 1}{n}},{n = 2},3\;,\;\ldots} & (26)\end{matrix}$

The corresponding values for Λ₁, b₂, and (2b₃−b₂) are

$\begin{matrix}{{\Lambda_{1} = \frac{2b_{3}}{{3n} - 1}},} & (27) \\{{b_{2} = {\left( {{2n} + 1} \right)\frac{\Lambda_{1}}{2}}},} & (28) \\{\left( {{2b_{3}} - b_{2}} \right) = {\left( {{4n} + 1} \right){\frac{\Lambda_{1}}{2}.}}} & (29)\end{matrix}$

Other values of η's that belong to the second set are given by theformulae

$\begin{matrix}{{\eta = \frac{{2\; n} \pm 1}{2n}},\;{n = 2},3,\;{\ldots\mspace{11mu}.}} & (30)\end{matrix}$

The corresponding values for Λ₁, b₂, and (2b₃−b₂) are

$\begin{matrix}{{\Lambda_{1} = \frac{2b_{3}}{{4n} \pm 1}},} & (31) \\{{b_{2} = {\left( {{2n} \pm 1} \right)\frac{\Lambda_{1}}{2}}},} & (32) \\{\left( {{2b_{3}} - b_{2}} \right) = {\left( {{6n} \pm 1} \right){\frac{\Lambda_{1}}{2}.}}} & (33)\end{matrix}$

The cut off spatial frequency in the determination of the surface figureproperties represented by ξ when using the three-axis/planeinterferometer will be determined by the spatial filtering properties ofmeasurement beams having a finite diameter, i.e., the finite size of ameasurement beam will serve as a low pass filter with respect to effectsof the higher frequency spatial frequencies of ξ. These effects areevaluated here for a measurement beam with a Gaussian profile. For aGaussian profile with a 1/e² diameter of 2s with respect to beamintensity, the effect of the spatial filtering of a Fourier seriescomponent with a spatial wavelength Λ is given by a transfer functionT_(Beam) where

$\begin{matrix}{T_{Beam} = {{\mathbb{e}}^{{- {(\frac{\pi^{2}}{8})}}{(\frac{2s}{\Lambda})}^{2}}.}} & (34)\end{matrix}$

Consider for example the effect of a Gaussian beam profile with a 1/e²diameter of 2s=5.0 mm. The attenuation effect of the spatial filteringwill be 0.225, 0.089, and 0.028 for Λ=4.55, 3.57, and 2.94 mm,respectively, for the respective Fourier series component amplitudes.

The series representations of SDP₁ ^(e) and SDP₂ ^(e) for the Fourierseries representation of ξ(y) given by Equation (13) are

$\begin{matrix}{{{SDP}_{1}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} +}} & (35) \\{\mspace{175mu}{{{\eta\frac{L}{2}\vartheta_{z}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}},}} & \; \\{\mspace{169mu}{{{\frac{1}{2}\left\lbrack {1 - {2\left( \frac{1}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack} \leq \frac{y}{L} \leq \frac{1}{2}},}} & \; \\{{{SDP}_{2}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (36) \\{\mspace{169mu}{{{\frac{L}{2}\vartheta_{z}} - {x\;\vartheta_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}}},}} & \; \\{\mspace{169mu}{{- \frac{1}{2}} \leq \frac{y}{L} \leq {- {{\frac{1}{2}\left\lbrack {1 - {2\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack}.}}}} & \;\end{matrix}$

Note in Equations (35) and (36) that the constant term A₀ is notpresent. The discussion of this feature relevant to the intrinsicsurface figure error function is the same as the corresponding portionof the discussion related to the constant term A₀ not being included inEquation (14).

From a comparison of Equations (35) and (36), another property of SDP₁^(e) and SDP₂ ^(e) is evident, i.e., the respective contributions of theFourier series terms in A′_(m) and B′_(m) are the same in SDP₁ ^(e) andSDP₂ ^(e). Furthermore, the respective contributions of the Fourierseries terms in A′_(m) and B′_(m) in SDP₁ ^(e) and SDP₂ ^(e) are thesame as the respective contributions in SDP (see Equation (16)).

The SDP₁ ^(e) and SDP₂ ^(e)are classified as conjugate other SDP relatedparameters because of the cited property and because the first orderterms θLθ_(z) and −Lη_(z), respectively, multiplied by the respectivedomains in y, i.e., b₃/(1+η) and ηb₃/(1+η), respectively, are equal inmagnitude and have opposite signs.

While certain SDP^(e)s have been explicitly defined, there are alsoother SDP^(e)s for L=4b₃+q2b₂+p2(b₃−b₂), q=1, 2, . . . , p=1, 2, . . . ,that exhibit similar properties as those of identified with respect toSDP₁ ^(e) and SDP₂ ^(e), such as an invariance to displacements of thestage mirror as evident on inspection of Equations (11) and (12).Another property of SDP₁ ^(e) and SDP₂ ^(e) is that the effect of astage rotation and the corresponding changes in SDP₁ ^(e) and SDP₂ ^(e)multiplied by their respective domain widths in y are equal in magnitudebut opposite in signs. This property is evident on examination ofEquations (35) and (36).

In some embodiments, the effects of the offset errors and the effects ofchanges in q, that occur during the measurements of x₁(y), x₂(y), andx₃(y) are next included in the representations of SDP, SDP₁ ^(e), andSDP₂ ^(e). The offset errors for x₁, x₂, and x₃ are represented by E₁,E₂, and E₃, respectively. Offset errors arise in SDP because SDP isderived from three different interferometer measurements where each ofthe three interferometers can only measure relative changes in arespective reference and measurement beam paths. In addition, the offseterrors may change with time because the calibrations of each of thethree interferometers may change with respect to each other, e.g. due tochanges in temperature. The effects of changes in the orientation of thestage during the measurements of SDP, SDP₁ ^(e), and SDP₂ ^(e) areaccommodated by representing the orientation of the stage θ_(z) asθ_(z)(y).

The representations of SDP, SDP₁ ^(e), and SDP₂ ^(e) that include theeffects of offset errors and changes in stage orientation areaccordingly expressed as

$\begin{matrix}{{{SDP}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (37) \\{\mspace{175mu}{{{x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}} - \left\lbrack {E_{1} - {\left( {1 + \eta} \right)E_{2}} + {\eta\; E_{3}}} \right\rbrack},}} & \; \\{\mspace{140mu}{{{- {\frac{1}{2}\left\lbrack {1 - {2\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack}} \leq \frac{y}{L} \leq {\frac{1}{2}\left\lbrack {1 - {2\left( \frac{1}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack}};}} & \; \\{{{SDP}_{1}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (38) \\{\mspace{191mu}{{x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}} - {\left( {1 + {\eta\; q}} \right)E_{1}} +}} & \; \\{\mspace{191mu}{{\left( {1 + \eta} \right)E_{2}} + {{\eta\left( {q - 1} \right)}E_{3}} +}} & \; \\{\mspace{194mu}{{\eta\; b_{3}\begin{Bmatrix}{{\vartheta_{z}(y)} + {\vartheta_{z}\left( {y - {2\; b_{3}}} \right)} + \ldots +} \\{\mspace{79mu}{\vartheta_{z}\left\lbrack {y - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}},}} & \; \\{\mspace{191mu}{{{\frac{1}{2}\left\lbrack {1 - {2\left( \frac{1}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack} \leq \frac{y}{L} \leq \frac{1}{2}};{and}}} & \; \\{{{SDP}_{2}^{e}\left( {x,y,\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (39) \\{\mspace{191mu}{{x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y,{\vartheta_{z} = 0}} \right)} \right\rbrack}} + {\left( {q - 1} \right)E_{1}} +}} & \; \\{\mspace{191mu}{{\left( {1 + \eta} \right)E_{2}} - {\left( {q + \eta} \right)E_{3}} +}} & \; \\{\mspace{191mu}{{b_{3}\begin{Bmatrix}{{\vartheta_{z}(y)} + {\vartheta_{z}\left( {y + L - {2\; b_{3}}} \right)} +} \\{{\vartheta_{z}\left( {y + L - {4b_{3}}} \right)} + \ldots +} \\{\mspace{31mu}{\vartheta_{z}\left\lbrack {y + L - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}},}} & \; \\{\mspace{191mu}{{- \frac{1}{2}} \leq \frac{y}{L} \leq {- {\frac{1}{2}\left\lbrack {1 - {2\left( \frac{\eta}{1 + \eta} \right)\left( \frac{2\; b_{3}}{L} \right)}} \right\rbrack}}}} & \;\end{matrix}$

-   -   where fourth order of effects arising from the y dependence of        θ_(z)(y) have been neglected in Equations (37), (38), and (39)        and θ _(z) represents the average value of θ_(z)(y) over the        domain in y.

The lower and upper limits y₁ and y₂, respectively, of the domains in yfor SDP₁ ^(e), and SDP₂ ^(e), respectively, are

$\begin{matrix}{{y_{1} = {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)}}},} & (40) \\{y_{2} = {y_{0} - \frac{L}{2} + {2\; b_{2}}}} & (41)\end{matrix}$

where y⁰ corresponds to the value of y at the middle of the domain in y.At the lower and upper limits y₁ and y₂, the respective SDP₁ ^(e) andSDP₂ ^(e) are expressed as

$\begin{matrix}{{{SDP}_{1}^{e}\left( {x,y_{1},\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y_{1}}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y_{1}}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (42) \\{\mspace{191mu}{{x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{1}^{e}\left( {y_{1},{\vartheta_{z} = 0}} \right)} \right\rbrack}} - {\left( {1 + {\eta\; q}} \right)E_{1}} +}} & \; \\{\mspace{191mu}{{\left( {1 + \eta} \right)E_{2}} + {{\eta\left( {q - 1} \right)}E_{3}} +}} & \; \\{\mspace{194mu}{{\eta\; b_{3}\begin{Bmatrix}{{{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)}} \right\rbrack} +}\mspace{121mu}} \\{{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)} - {2b_{3}}} \right\rbrack} + \ldots +} \\{\;{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} - {2\left( {b_{3} - b_{2}} \right)} - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}},}} & \; \\{{{SDP}_{2}^{e}\left( {x,y_{2},\vartheta_{z}} \right)} = {{\frac{1}{2}{\sum\limits_{m = 1}^{N}\left\lbrack {{{\cos\left( {m\; 2\;\pi\frac{y_{2}}{L}} \right)}A_{m}^{\prime}} + {{\sin\left( {m\; 2\;\pi\frac{y_{2}}{L}} \right)}B_{m}^{\prime}}} \right\rbrack}} -}} & (43) \\{\mspace{191mu}{{x\;{\overset{\_}{\vartheta}}_{z}{\frac{\partial}{\partial y}\left\lbrack {{SDP}_{2}^{e}\left( {y_{2},{\vartheta_{z} = 0}} \right)} \right\rbrack}} + {\left( {q - 1} \right)E_{1}} +}} & \; \\{\mspace{191mu}{{\left( {1 + \eta} \right)E_{2}} - {\left( {q + \eta} \right)E_{3}} +}} & \; \\{\mspace{194mu}{b_{3}{\begin{Bmatrix}{{{\vartheta_{z}\left\lbrack {y_{0} - \frac{L}{2} + {2b_{2}}} \right\rbrack} +}\mspace{121mu}} \\{{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right\rbrack} +} \\{{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {4b_{3}}} \right\rbrack} + \ldots +} \\{\;{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}}\end{Bmatrix}.}}} & \;\end{matrix}$

It is evident on examination of Equations (37), (42), and (43) that thecontribution of the surface figure error function terms to SDP and SDP₁^(e) are continuous at y₁ and that the contribution of the surfacefigure error function terms to SDP and SDP₂ ^(e) are continuous at y₂.This is a very significant property which is subsequently used in aprocedure to eliminate the effects of offset errors E₁, E₂, and E₃ andthe effects of the y dependence of θ_(z)(y).

As a consequence of the property, the differences between SDP and SDP₁^(e) at y₁ and between SDP and SDP₂ ^(e) at y₂ are independent of thesurface figure error function except for the values of the surfacefigure error at the extreme edges of the domain being mapped.Expressions for the differences are obtained using Equations (37), (42),and (43) with the results

$\begin{matrix}{\left\lbrack {{{SDP}_{1}^{e}\left( {x,y_{1},\vartheta_{z}} \right)} - {{SDP}\left( y_{1} \right)}} \right\rbrack = {{{- \eta}\;{q\left( {E_{1} - E_{3}} \right)}} +}} & (44) \\{\mspace{355mu}{\left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right) +}} & \; \\{\mspace{329mu}{{\eta\; b_{3}\begin{Bmatrix}{{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right)} +}\mspace{56mu}} \\{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {4b_{3}}} \right)} + \ldots +} \\{\mspace{59mu}{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2q\; b_{3}}} \right)}}\end{Bmatrix}},}} & \; \\{\left\lbrack {{{SDP}\left( y_{2} \right)} - {{SDP}_{2}^{e}\left( {x,y_{2},\vartheta_{z}} \right)}} \right\rbrack = {{- {q\left( {E_{1} - E_{3}} \right)}} +}} & (45) \\{\mspace{355mu}{\left( \frac{{\xi_{3}\left( {y_{0} + \frac{L}{2}} \right)} - {\xi_{1}\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right) +}} & \; \\{\mspace{335mu}{b_{3}{\begin{Bmatrix}{{{\vartheta_{z}\left( {y_{0} - \frac{L}{2} + {2b_{2}}} \right)} +}\mspace{121mu}} \\{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right)} +} \\{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {4\; b_{3}}} \right)} + \ldots +} \\{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2\left( {q - 1} \right)b_{3}}} \right\rbrack}\end{Bmatrix}.}}} & \;\end{matrix}$

Using the relationship L=2qb₃ given by Equation (8), Equation (45) iswritten as

$\begin{matrix}{\left\lbrack {{{SDP}\left( y_{2} \right)} - {{SDP}_{2}^{e}\left( {x,y_{2},\vartheta_{z}} \right)}} \right\rbrack = {{- {q\left( {E_{1} - E_{3}} \right)}} +}} & (46) \\{\mspace{340mu}{\left( \frac{{\xi\left( {y_{0} + \frac{L}{2}} \right)} - {\xi_{1}\left( {y_{0} - \frac{L}{2}} \right)}}{2} \right) +}} & \; \\{\mspace{335mu}{b_{3}{\begin{Bmatrix}{{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {2b_{3}}} \right)} +}\mspace{56mu}} \\{{\vartheta_{z}\left( {y_{0} + \frac{L}{2} + {2b_{2}} - {4b_{3}}} \right)} + \ldots +} \\{\mspace{59mu}{\vartheta_{z}\left\lbrack {y_{0} + \frac{L}{2} + {2b_{2}} - {2q\; b_{3}}} \right\rbrack}}\end{Bmatrix}.}}} & \;\end{matrix}$

The ratio of the value of the discontinuity D₁(y₀) given Equation (44)and the value of the discontinuity D₂ (y₀) given by Equation (46) isthus equal to η, i.e.D ₁(y ₀)=ηD ₂(y ₀)  (47)

whereD ₁(y ₀)≡[SDP ₁ ^(e)(x, y ₁,θ_(z))−SDP(y ₁,θ_(z))],  (48)D ₂(y ₀)≡[SDP(y ₂,θ_(z))−SDP ₂ ^(e)(x, y ₂,θ_(z))].  (49)

Based on the foregoing mathematical development, a general procedure fordetermining a surface figure error function, ξ(y), without any priorknowledge of the surface figure is as follows. First, select the lengthof the mirror that is to be mapped by selecting a value of q. The lengthof the mirror is given by Equation (8). Orient the stage mirror so thatθ_(z) is at or about zero and the distance x from the interferometer tothe stage mirror being mapped is relatively small. The small distancecan reduce the contribution to the geometric error correction term.While maintaining a nominal x distance to the stage mirror, acquiresimultaneous values for x₁, x₂, and x₃ while scanning the mirror in they-direction and monitor the position of the mirror in the y-directionand changes in stage orientation during the scan. The x₁, x₂, and x₃values, the position of the mirror in the y-direction, and the changesin stage orientation can be stored to the electronic controller's memoryor to disk. The stored data will be in the form of a 3×N array for thex₁, x₂, and x₃ information, where N is the total number of samplesacross the mirror, a 1×N array for y position information and a 1×Narray for changes in stage orientation information.

Next, calculate values for SDP, SDP₁ ^(e), and SDP₂ ^(e) on a samplinggrid (e.g., a uniform sampling grid) in preparation for a discreteFourier transform which will be performed later. These calculations caninclude averaging multiple values of SDP, SDP₁ ^(e), and SDP₂ ^(e) whereeach increment in the sampling grid corresponds to multiple values ofx₁, x₂, and x₃. Since the SDP₁ ^(e), and SDP₂ ^(e) parameters aresensitive to changes in stage orientation during the scan, the arraycontaining changes in stage orientation information during the scan isused to correct the SDP₁ ^(e) and SDP₂ ^(e) parameters to a chosenreference stage orientation.

Next, remove the discontinuities at the domain boundaries between SDP₁^(e) and SDP, and between SDP and SDP₂ ^(e) by adding appropriateconstant values to SDP₁ ^(e) and SDP₂ ^(e). These constant values are−D₁ and D₂ as given by Equations (48) and (49).

After the discontinuities have been removed, a resultant SDP array isconstructed by concatenating the SDP′₂, SDP and SDP₁ ^(e) arrays. Theresultant SDP array spans the length of the mirror to be mapped, L.Next, perform a discrete Fourier transform on the resultant SDP array.Using a complex transfer function calculated for the interferometerbeing used, transform the SDP Fourier coefficients into the Fouriercoefficients for the surface figure error function ξ(y). The surfacefigure error function, ξ(y), can now be determined from the Fouriercoefficients using Equation (13).

In some embodiments, this procedure can be used to determine surfacefigure error functions for mirrors in a lithography tool at times whenthe tool is not being used to expose wafers. For example, this procedurecan be used during routine maintenance of the tool. At such times, adedicated scan sequence can be implemented, allowing a portion of amirror to be scanned at a rate optimized for mirror characterizationpurposes. Moreover, multiple scans can be performed for differentrelative distances between the mirror and the interferometer and atdifferent nominal stage orientations, providing a family of ultimatesurface figure error functions which can be interpolated for use atother relative distances between the mirror and the interferometer andat other stage orientations.

In embodiments where data is acquired during dedicated scan sequences,the data can be acquired over time periods that are relatively long withrespect to various sources of random errors in x₁, x₂, and x₃measurements. Accordingly, uncertainty in surface figure error functionsdue to random errors sources can be reduced. For example, one source ofrandom errors are fluctuations in the composition or density of theatmosphere in the interferometer beam paths. Over relatively long timeperiods, these fluctuations average to zero. Accordingly, acquiring datafor calculating SDP and SDP^(e) values over sufficiently long periodscan substantially reduce the uncertainty of ξ(y) due to thesenon-stationary atmospheric fluctuations.

In certain embodiments where the mirror is to be mapped in situ during,for example, lithography tool operation, arrays of x₁, x₂, and x₃, yposition and stage orientation can be accuired and stored to theelectronic controller's memory or to disk for specific relativedistances and nominal stage orientations. After the arrays of data havebeen acquired, the remaining steps for mapping the surface figure errorfunction are the same as in an offline analysis. Values for SDP, SDP₁^(e), and SDP₂ ^(e) can be calculated and assigned to a sampling grid(e.g., a uniform sampling grid) in preparation for a discrete Fouriertransform which will be performed later. These calculations can includeaveraging multiple values of SDP, SDP₁ ^(e), and SDP₂ ^(e) where eachincrement in the sampling grid corresponds to multiple values of x₁, x₂,and x₃. Since the SDP₁ ^(e), and SDP₂ ^(e) parameters are sensitive tochanges in stage orientation during the scan, the array containingchanges in stage orientation information during the scan is used tocorrect the SDP₁ ^(e) and SDP₂ ^(e) parameters to a chosen referencestage orientation.

Next, the discontinuities at the domain boundaries between SDP₁ ^(e) andSDP, and between SDP and SDP₂ ^(e) are removed by adding appropriateconstant values to SDP₁ ^(e) and SDP₂ ^(e). These constant values are−D₁ and D₂ as given by Equations (48) and (49).

After the discontinuities have been removed, a resultant SDP array isconstructed by appropriately concatenating the SDP₂ ^(e), SDP and SDP₁^(e) arrays. The resultant SDP array spans the length of the mirror tobe mapped, L. Next, perform a discrete Fourier transform on theresultant SDP array. Using a complex transfer function calculated forthe interferometer being used, transform the SDP Fourier coefficientsinto the Fourier coefficients for the surface figure error functionξ(y). The surface figure error function, ξ(y), can now be determinedfrom the Fourier coefficients using Equation (13).

The surface figure error function may be updated and maintained bycomputing a running average of the surface figure error functionacquired at successive time intervals.

The running average can be stored to the electronic controller's memoryor to disk. The optimum time interval which a given average spans can bechosen after determining the time scales over which the surface figureerror function ξ(y) significantly changes.

In certain other embodiments, a predetermined surface figure errorfunction ξ can be updated or variations in ξ can be monitored based onvalues of SDP and/or SDP^(e) s calculated from newly-acquiredinterferometry data. For example, in a lithography tool, x₁, x₂, and x₃values acquired during operation of the tool (e.g., during waferexposure) can be used to update a mirror's surface figure errorfunction, or can be used to monitor changes in the surface figure errorfunction and provide an indicator of when new full mirrorcharacterizations are required to maintain the interferometry system'saccuracy.

In implementations where the surface figure error function is updatedduring an exposure sequence of a tool, data can be acquired overmultiple wafer exposure sequences so that changes that occur to thesurface figure error function as a result of systematic changes in thetool that occur each sequence can be distinguished from actual permanentchanges in the mirror's characteristics. One example of systematicchanges in a tool that may contribute to variations a surface figureerror function are stationary changes in the atmosphere in theinterferometer's beam paths. Such changes occur for each wafer exposurecycle as the composition and/or gas pressure inside the lithography toolis changed. In the absence of any other technique for monitoring and/orcompensating for these effects, they can result in errors in the x1, x2,and x3 measurements that ultimately manifest as errors in the surfacefigure error function. However, since the changes to the atmosphere arestationary, their effect on ξ(y) can be reduced (e.g., eliminated) byidentifying variations that occur to ξ(y) with the same period as thewafer exposure cycle, and subtracting these variations from thecorrection that is ultimately made to a stage measurement.

The predetermined surface figure error function in the certain otherembodiments may be updated by creating a running average of the surfacefigure error function using the predetermined surface figure errorfunction and surface figure error functions acquired in situ atsubsequent time intervals. The subsequent surface figure error functionsacquired in situ may be acquired as in the certain embodimentsdescription above. The running average can be stored to the electroniccontroller's memory or to disk. The optimum time interval which a givenaverage spans can be chosen after determining the time scales over whichthe surface figure error function ξ(y) significantly changes.

An alternative procedure for updating a predetermined surface figure inthe certain other embodiments above may be used. The first step in thealternate procedure is to measure values of SDP(y, θ _(z)=0) for asection of the x-axis stage mirror. The length of the section or domainis ≧4b₃ and can include the entire length of the stage mirror. Next,integrals of SDP(y, θ _(z)=0) with weight functions cos(s2πy/L) andsin(s2πy/L) are computed from the measured values of SDP(y, θ _(z)=0)minus the mean value <SDP(y, θ _(z)=0)> of SDP(y, θ _(z)=0), SDP₁ ^(e)(y, θ _(z)=0)−ηD(y₀), and SDP₂ ^(e)(y, θ _(z)=0)+D(y₀), <SDP(y, θ_(z)=0)>. The integrals are:

$\begin{matrix}{A_{s}^{''} = {\frac{4}{\left( {L - {2b_{3}}} \right)} \times {\int_{({{- \frac{L}{2}} + {2b_{2}}})}^{\lbrack{\frac{L}{2} - {2{({b_{3} - b_{2}})}}}\rbrack}\left\lbrack {{{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z}\  = 0}} \right)} -} \right.}}} & (50) \\{{\left. \mspace{259mu}\left\langle {{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z}\  = 0}} \right)} \right\rangle \right\rbrack{\cos\left( \frac{{s2}\;\pi\; y}{L} \right)}{\mathbb{d}y}},\;{s \geq 1},} & \; \\{B_{s}^{''} = {\frac{4}{\left( {L - {2b_{3}}} \right)} \times {\int_{({{- \frac{L}{2}} + {2b_{2}}})}^{\lbrack{\frac{L}{2} - {2{({b_{3} - b_{2}})}}}\rbrack}\left\lbrack {{{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z}\  = 0}} \right)} -} \right.}}} & (51) \\{{\left. \mspace{259mu}\left\langle {{SDP}\left( {y,{{\overset{\_}{\vartheta}}_{z}\  = 0}} \right)} \right\rangle \right\rbrack{\sin\left( \frac{{s2}\;\pi\; y}{L} \right)}{\mathbb{d}y}},\;{s \geq 1.}} & \;\end{matrix}$

The integrals expressed by Equations (50) and (51) are evaluated usingthe series representation of SDP given by Equation (16) with the results

$\begin{matrix}{{A_{q}^{''} = {\frac{4}{\left( {L - {2b_{3}}} \right)} \times {\int_{({{- \frac{L}{2}} + {2b_{2}}})}^{\lbrack{\frac{L}{2} - {2{({b_{3} - b_{2}})}}}\rbrack}{\begin{bmatrix}{{\sum\limits_{m = 1}^{N}\;{A_{m}^{\prime}{\cos\left( \frac{{m2}\;\pi\; y}{L} \right)}{\cos\left( \frac{q\; 2\;\pi\; y}{L} \right)}}} +} \\{\mspace{25mu}{\sum\limits_{m = 1}^{N}\;{B_{m}^{\prime}{\sin\left( \frac{{m2}\;\pi\; y}{L} \right)}{\cos\left( \frac{q\; 2\;\pi\; y}{L} \right)}}}}\end{bmatrix}{\mathbb{d}y}}}}},{q \geq 1},} & (52) \\{{B_{q}^{''} = {\frac{4}{\left( {L - {2b_{3}}} \right)} \times {\int_{({{- \frac{L}{2}} + {2b_{2}}})}^{\lbrack{\frac{L}{2} - {2{({b_{3} - b_{2}})}}}\rbrack}{\begin{bmatrix}{{\sum\limits_{m = 1}^{N}\;{A_{m}^{\prime}{\cos\left( \frac{{m2}\;\pi\; y}{L} \right)}{\cos\left( \frac{q\; 2\;\pi\; y}{L} \right)}}} +} \\{\mspace{25mu}{\sum\limits_{m = 1}^{N}\;{B_{m}^{\prime}{\sin\left( \frac{{m2}\;\pi\; y}{L} \right)}{\cos\left( \frac{q\; 2\;\pi\; y}{L} \right)}}}}\end{bmatrix}{\mathbb{d}y}}}}},{q \geq 1.}} & (53)\end{matrix}$

where coefficients A′_(q) and B′_(q) are with respect to SDP(y, θ_(z)=0) with <SDP(y, θ _(z)=0)> subtracted. The evaluation of theintegrals in Equation (52) is next performed with the results

$\begin{matrix}{{A_{q}^{\prime\prime} = {\frac{1}{\left( {1 - \frac{2b_{3}}{L}} \right)} \times \begin{Bmatrix}{\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\frac{1}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\sin\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{+ \sin}\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} \\{{+ \frac{1}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{+ \sin}\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}} \\{- {\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{\frac{1}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{- \cos}\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} \\{{- \frac{1}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{- \cos}\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}}}\end{Bmatrix}}},\mspace{11mu}{q \geq 1},} & (54) \\{{{A_{q}^{\prime\prime} = {A_{q}^{\prime} + {\frac{1}{\left( {1 - \frac{2b_{3}}{L}} \right)} \times \begin{Bmatrix}{\sum\limits_{\substack{{m = 1}, \\ m \neq q}}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\sin\;{{\pi\left( {q + m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{+ \sin}\;{\pi\left( {q + m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\end{Bmatrix}} \\{{+ \frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {4\;\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{+ \sin}\;{\pi\left( {q - m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\end{Bmatrix}}\end{pmatrix} \right)}} \\{- {\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{\frac{\left( {- 1} \right)^{q - m}}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{- \cos}\;{{\pi\left( {q + m} \right)}\left\lbrack {4\;\frac{b_{2}}{L}} \right\rbrack}}\end{Bmatrix}} \\{{- \frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{- \cos}\;{\pi\left( {q - m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\end{Bmatrix}}\end{pmatrix} \right)}}}\end{Bmatrix}}}},{q \geq 1},}\mspace{25mu}} & (55) \\{{A_{q}^{\prime\prime} = {A_{q}^{\prime} + {\frac{1}{\left( {1 - \frac{2b_{3}}{L}} \right)} \times \left\{ \begin{matrix}{\sum\limits_{\substack{{m = 1}, \\ m \neq q}}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\sin\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{+ {\sin\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}} \\{{+ \frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\sin\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{+ {\sin\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}}\end{pmatrix} \right)}} \\{- {\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{\frac{\left( {- 1} \right)^{q - m}}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\cos\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{- {\cos\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}} \\{{- \frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\cos\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{- {\cos\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}}\end{pmatrix} \right)}}}\end{matrix} \right\}}}},{q \geq 1},} & (56) \\{{{A_{q}^{\prime\prime} = {A_{q}^{\prime} + {\frac{\left( \frac{2b_{3}}{L} \right)}{\left\lbrack {1 - \left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times \begin{Bmatrix}{{\sum\limits_{\substack{{m = 1}, \\ m \neq q}}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\left( {- 1} \right)^{q + m + 1}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q + m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\cos\left\lbrack {{\pi\left( {q + m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}}\end{Bmatrix}} \\{{+ \left( {- 1} \right)^{q - m + 1}}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\cos\left\lbrack {{\pi\left( {q - m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}} +} \\{\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{\left( {- 1} \right)^{q - m + 1}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q + m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\sin\left\lbrack {{\pi\left( {q + m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}}\end{Bmatrix}} \\{{- \left( {- 1} \right)^{q - m + 1}}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\sin\left\lbrack {{\pi\left( {q - m} \right)}\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}}\end{Bmatrix}}}},{q \geq 1.}}\;} & (57)\end{matrix}$

The evaluation of the integrals in Equation (53) is next performed withthe results

$\begin{matrix}{B_{q}^{\prime\prime} = {\frac{1}{\left( {1 - \frac{2b_{3}}{L}} \right)} \times}} & (58) \\{\left\{ \begin{matrix}{- {\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\frac{1}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{- \cos}\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} \\{{+ \frac{1}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{- \cos}\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}}} \\{+ {\sum\limits_{m = 1}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{- \frac{1}{\left( {q + m} \right)2\pi}}\begin{Bmatrix}{\sin\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{+ \sin}\;{{\pi\left( {q + m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}} \\{{+ \frac{1}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\frac{\left( {b_{3} - b_{2}} \right)}{L}}} \right\rbrack}} \\{{+ \sin}\;{{\pi\left( {q - m} \right)}\left\lbrack {1 - {4\;\frac{b_{2}}{L}}} \right\rbrack}}\end{Bmatrix}}\end{pmatrix} \right)}}}\end{matrix} \right\},{q \geq 1},} & \; \\{B_{q}^{\prime\prime} = {B_{q}^{\prime} + {\frac{1}{\left( {1 - \frac{2b_{3}}{L}} \right)} \times}}} & (59) \\{\left\{ \begin{matrix}{- {\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\frac{\left( {- 1} \right)^{q + m}}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\cos\;{{\pi\left( {q + m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{- \cos}\;{{\pi\left( {q + m} \right)}\left\lbrack {4\;\frac{b_{2}}{L}} \right\rbrack}}\end{Bmatrix}} \\{{+ \frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\cos\;{{\pi\left( {q - m} \right)}\left\lbrack {4\;\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{- \cos}\;{\pi\left( {q - m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\end{Bmatrix}}\end{pmatrix} \right)}}} \\{+ {\sum\limits_{\underset{m \neq q}{{m = 1},}}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{- \frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)2\pi}}\begin{Bmatrix}{\sin\;{{\pi\left( {q + m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{+ \sin}\;{\pi\left( {q + m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\end{Bmatrix}} \\{{+ \frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\sin\;{{\pi\left( {q - m} \right)}\left\lbrack {4\frac{\left( {b_{3} - b_{2}} \right)}{L}} \right\rbrack}} \\{{+ \sin}\;{\pi\left( {q - m} \right)}\left( {4\;\frac{b_{2}}{L}} \right)}\end{Bmatrix}}\end{pmatrix} \right)}}}\end{matrix} \right\},{q \geq 1}} & \; \\{B_{q}^{\prime\prime} = {B_{q}^{\prime} + {\frac{1}{\left( {1 - \frac{2b_{3}}{L}} \right)} \times}}} & (60) \\{\left\{ \begin{matrix}{- {\sum\limits_{m = 1}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\frac{\left( {- 1} \right)^{q + m}}{\left( {q + m} \right)2\pi}\begin{Bmatrix}{\cos\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{- {\cos\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}} \\{{+ \frac{\left( {- 1} \right)^{q - m}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\cos\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{- {\cos\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}}\end{pmatrix} \right)}}} \\{+ {\sum\limits_{\substack{{m = 1}, \\ m \neq q}}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{- \frac{\left( {- 1} \right)^{q + m + 1}}{\left( {q + m} \right)2\pi}}\begin{Bmatrix}{\sin\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{+ {\sin\begin{bmatrix}{2{\pi\left( {q + m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}} \\{{+ \frac{\left( {- 1} \right)^{q - m + 1}}{\left( {q - m} \right)2\pi}}\begin{Bmatrix}{\sin\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{1}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}} \\{+ {\sin\begin{bmatrix}{2{\pi\left( {q - m} \right)}} \\{\frac{\eta}{\left( {1 + \eta} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}}\end{pmatrix} \right)}}}\end{matrix} \right\},{q \geq 1}} & \; \\{B_{q}^{\prime\prime} = {B_{q}^{\prime} + {\frac{\left( \frac{2b_{3}}{L} \right)}{\left\lbrack {1 - \left( \frac{2b_{3}}{L} \right)} \right\rbrack} \times}}} & (61) \\{\left\{ \begin{matrix}{\sum\limits_{{m = 1},}^{N}{A_{m}^{\prime}\left( \begin{pmatrix}{\left( {- 1} \right)^{q + m + 1}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q + m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\sin\begin{bmatrix}{\pi\left( {q + m} \right)} \\{\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}} \\{{+ \left( {- 1} \right)^{q - m + 1}}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\sin\begin{bmatrix}{\pi\left( {q - m} \right)} \\{\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}}\end{pmatrix} \right)}} \\{+ {\sum\limits_{\substack{{m = 1}, \\ m \neq q}}^{N}{B_{m}^{\prime}\left( \begin{pmatrix}{{- \left( {- 1} \right)^{q + m + 1}}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q + m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\cos\begin{bmatrix}{\pi\left( {q + m} \right)} \\{\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}} \\{{+ \left( {- 1} \right)^{q - m + 1}}\begin{Bmatrix}{{sinc}\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack} \\{\times {\cos\begin{bmatrix}{\pi\left( {q - m} \right)} \\{\frac{\left( {\eta - 1} \right)}{\left( {\eta + 1} \right)}\left( \frac{2b_{3}}{L} \right)}\end{bmatrix}}}\end{Bmatrix}}\end{pmatrix} \right)}}}\end{matrix} \right\},{q \geq 1.}} & \;\end{matrix}$

The effects of non-orthogonality of functions cos(m2πy/L) andsin(m2πy/L) over the domain of y in SDP(y, θ _(z)=0) are evident uponexamination of Equations (57) and (61). The effects of non-orthogonalityare represented by the differences (A″_(q)−A′_(q)) and (B″_(q)−B′_(q))in Equations (57) and (61), respectively.

Equations (57) and (61) can be solved for the A′_(q) and the B_(q)′,respectively, by an iterative procedure. The degree to shich the effectsof non-orthogonality are decoupled from the leading A′_(q) and B′_(q)terms in Equations (57) and (61) depends on the ratio (2b₃/L. The largercoefficients of “off diagonal terms” in Equations (57) and (61)generally occur for small values |q−m| and the relative magnitude r ofthe coefficients of these terms is

$\begin{matrix}{r \cong {\frac{\left( \frac{2b_{3}}{L} \right)}{1 - \left( \frac{2b_{3}}{L} \right)}{{{sinc}\left\lbrack {{\pi\left( {q - m} \right)}\left( \frac{2b_{3}}{L} \right)} \right\rbrack}.}}} & (62)\end{matrix}$

The iterative procedure is a nontrivial step for which two solutions aregiven. The step is nontrivial since cos(m2πy/L) and sin(m2πy/L) arefunctions that are not orthogonal as a set over the domain ofintegration in y. One solution involves using a surface figure errorfunction ξ obtained at an earlier time based on measured values of SDP,SDP₁ ^(e), and SDP₂ ^(e) to compute values for the off-diagonal terms inEquations (57) and (61).

Alternatively, or additionally, information about a surface figure errorfunction can be obtained using other techniques, such as based on valuesof a FDP such as given by Equation (3) or from interferometricmeasurements made ex-situ, e.g., a Fizeau interferometer, before thestage mirror is installed in a lithography tool.

The resulting values for A′_(m) and B′_(m) may be used without anyiteration if the resulting values for A′_(m) and B′_(m) indicate thatthere has not been any significant change in the surface figure errorfunction ξ. If there has been a significant change, the resulting valuesfor A′_(m) and B′_(m) are used as input information to compute theoff-diagonal terms in a second step of the iterative procedure. Theiterative steps are repeated until stable, i.e., asymptotic, values forA′_(m) and B′_(m) are obtained. The values for A_(m) and B_(m) aresubsequently obtained using Equations (17) and (18) from thecoefficients A′_(m) and B′_(m) produced by the iterative procedure.

After the surface figure error function ξ(y,θ_(z)=0) is determined forthe specified section of the x-axis stage mirror, the procedure can berepeated for other sections of the x-axis stage mirror that are used inthe respective lithography tool.

In some embodiments, measured values of SDP can be used to extend aknown surface figure error function ξ to other sections of the x-axisstage mirror. Equation (7) allows the generation of figure errorfunction ξ(y,θ_(z)=0) for the other sections of the x-axis stage mirrorfrom the known surface figure error function ξ(y,θ_(z)=0) in an adjacentsection of the x-axis stage mirror.

In certain embodiments, local spatial derivatives of a surface figureerror function can be measured using the methods disclosed herein, insome cases, with relatively high sensitivity. The sensitivity of thethird order term represented by the last term in Equation (16) isproportional to m and is equal, for example, to the sensitivity of theremaining first order terms in Equation (16) for m≈100 for |θ_(z)|=0.5milliradians and x=0.7. To measure the local spatial derivativesdirectly or with a higher sensitivity, the procedure described for thedetermination of ξ(y,θ_(z)=0) can be repeated for non-zero values ofθ_(z) and for large values of x and y for the x-axis and y-axis stagemirrors, respectively.

The surface figure error function ξ(y, θ _(z)=0) obtained in the abovedescribed procedures includes in particular information about ξ(y, θ_(z)=0) that is quadratic in y. However, the surface figure errorfunction ξ(y, θ _(z)=0) obtained in the above described procedure doesnot represent any error in ξ(y, θ _(z)=0) that is linear in y since SDPis zero for a plane mirror surface.

The error in the determined ξ(y, θ _(z)=0) linear in y relative to theerror in the determined ξ(y, θ _(z)=0) linear in x for the x-axis andthe y-axis stage mirrors, respectively, i.e., a differential error, isthe only portion of the linear error that is relevant. A common modeerror corresponds to a rotation of the stage. The differential error isrelated to the angle between the x-axis and y-axis stage mirrors and theeffect of the differential error is compensated by measuring the anglebetween the two stage mirrors. The angle between the two stage mirrorscan be measured by use of an artifact wafer and the artifact waferrotated by 90 degrees for each plane defined by the multiple-axes/planeinterferometers.

It is the information about the angles between the two stage mirrorsthat is used for compensating surface figure error functions in yaw.Additional information about the respective surface figure errorfunctions may also be obtained by introducing a displacement of thesecond pass measurement beams at the stage mirror by changing theorientation of the stage mirror to change the respective pitch andrepeating the procedures described herein to determine the surfacefigure error functions at the position of the sheared measurement beams.

If the mean value <SDP(y, θ _(z)=0)> is not subtracted from measuredvalues of SDP(y, θ _(z)=0), information about the quadratic term in therepresentation of the surface figure error function may be lost. This isbecause the mean value of the measured values of SDP(y, θ _(z)=0)represents an error in 4 (y, θ _(z)=0) that is quadratic in y and anoffset error due to offset errors in the linear displacements x₁, x₂,and x₃. As a consequence, the error in the determined ξ( θ _(z)=0) thatis quadratic in y is measured by an independent procedure such as use ofan artifact wafer and the artifact wafer rotated by 180 degrees.

Values of θ_(z)(y) are measured by the y-axis interferometer. During ascan of the x-axis mirror, typically there is nominally no change in thelocation stage in the x direction. Accordingly, there is no change inthe nominal location of the measurement beams of the y-axisinterferometer on the surface of the corresponding y-axis stage mirror.However, the relative scales of the measurement axes of the y-axisinterferometer used to measure θ_(z)(y) will not, in general, beidentical and furthermore may be a function of y. The respective scalesmay be different due to geometric effects such as described in commonlyowned U.S. patent application Ser. No. 10/872,304 entitled “COMPENSATIONFOR GEOMETRIC EFFECTS OF BEAM MISALIGNMENTS IN PLANE MIRRORINTERFEROMETER METROLOGY SYSTEMS” or due to stationary systematicchanges in the density gradients of the gas in the y-axis interferometermeasurement beam paths.

Such errors due to relative scale errors can be difficult to measure andare typically done using an artifact wafer. However, measured values ofthe SDP and values of SDP reconstructed from the derived mirror surfacefigure error function can be used to test to see if such errors arepresent and if so, measure the y dependence of the relative scale errorsthat are quadratic and higher order in terms of y. Over time scalesshort compared to the time scales for changes in (E₁−E₃), the differencebetween measured values of SDP and values of SDP reconstructed from thederived mirror surface figure error function are obtained as a functionof y. Relative scaling errors in the y axes interferometer measurementswhich result in θ_(z)(y) having a quadratic error term with respect to ycan be identified as a term linear in y for the difference betweenmeasured and reconstructed values of SDP. Similarly errors in θ_(z)(y)which are cubic and higher order with respect to y can be identified asquadratic and higher order terms in y for the difference betweenmeasured and reconstructed values of SDP.

The procedure described for determining and compensating errors in themeasured values of θ_(z)(y) also include compensating the effects ofstationary effects of the gas in the respective measurement andreference beam paths

The surface figure error function for an x-axis stage mirror will, ingeneral, be a function of both y and z and the surface figure errorfunction for the y-axis stage mirror will in general be a function ofboth x and z. The generalization to cover the z dependent properties issubsequently addressed by the use of a two three-axis/planeinterferometer system in conjunction with a procedure to measure theangle between the x-axis and y-axis stage mirrors in two differentparallel planes corresponding to the two planes defined by the twothree-axis/plane interferometer system.

Once determined, surface figure error functions are typically used tocompensate interferometry system measurements, thereby improving aninterferometry system's accuracy. For example, ξ(y) can be used tocompensate measurements of the position of an x-axis stage mirror along,e.g., axis x₁ for a position y based on the following equations. If d isthe true x-axis displacement of the stage mirror and ξ′(y) is the truesurface figure error function, an uncorrected measurement at position y,will give, for example,

$\begin{matrix}{{{{\overset{\sim}{x}}_{1}(y)} = {d + \left( \frac{{\xi_{1}^{\prime}(y)} + {\xi_{0}^{\prime}(y)}}{2} \right) + {d_{nom}{\theta_{z}\left( {\frac{\partial\xi_{1}^{\prime}}{\partial y} - \frac{\partial\xi_{0}^{\prime}}{\,_{3}{\partial y}}} \right)}}}},} & (63)\end{matrix}$

where ξ′₁ and ξ′₀ refer to the surface figure error function values atpositions x′₁ and x′₀, respectively, and d_(nom) is a nominaldisplacement value used to calculate the geometric error term thatoccurs for non-zero θ_(z).

Using the surface figure error function, a compensated displacementx_(c)(y) can be calculated from the measured displacement based on thefollowing equation:

$\begin{matrix}{{x_{c}(y)} = {{{\overset{\sim}{x}}_{1}(y)} - \left( \frac{{\xi_{1}(y)} + {\xi_{0}(y)}}{2} \right) - {d_{nom}{{\theta_{z}\left( {\frac{\partial\xi_{1}}{\partial y} - \frac{\partial\xi_{0}}{\,_{1}{\partial y}}} \right)}.}}}} & (64)\end{matrix}$

In embodiments, various other error compensation techniques can be usedto reduce other sources of error in interferometer measurements. Forexample, cyclic errors that are present in the linear displacementmeasurements can be reduced (e.g., eliminated) and/or compensated by useof one of more techniques such as described in commonly owned U.S.patent application Ser. No. 10/097,365, entitled “CYCLIC ERROR REDUCTIONIN AVERAGE INTERFEROMETRIC MEASUREMENTS,” and U.S. patent applicationSer. No. 10/616,504 entitled “CYCLIC ERROR COMPENSATION ININTERFEROMETRY SYSTEMS,” which claims priority to Provisional PatentApplication No. 60/394,418 entitled “ELECTRONIC CYCLIC ERRORCOMPENSATION FOR LOW SLEW RATES,” all of which are by Henry A. Hill andthe contents of which are incorporated herein in their entirety byreference.

An example of another cyclic error compensation technique is describedin commonly owned U.S. patent application Ser. No. 10/287,898 entitled“INTERFEROMETRIC CYCLIC ERROR COMPENSATION,” which claims priority toProvisional Patent Application No. 60/337,478 entitled “CYCLIC ERRORCOMPENSATION AND RESOLUTION ENHANCEMENT,” by Henry A. Hill, the contentsof which are incorporated herein in their entirety by reference.

Another example of a cyclic error compensation technique is described inU.S. patent application Ser. No. 10/174,149 entitled “INTERFEROMETRYSYSTEM AND METHOD EMPLOYING AN ANGULAR DIFFERENCE IN PROPAGATION BETWEENORTHOGONALLY POLARIZED INPUT BEAM COMPONENTS,” which claims priority toProvisional Patent Application No. 60/303,299 entitled “INTERFEROMETRYSYSTEM AND METHOD EMPLOYING AN ANGULAR DIFFERENCE IN PROPAGATION BETWEENORTHOGONALLY POLARIZED INPUT BEAM COMPONENTS,” both by Henry A. Hill andPeter de Groot, the contents both of which are incorporated herein intheir entirety by reference.

A further example of a cyclic error compensation technique is describedin commonly owned Provisional Patent Application No. 60/314,490 filedentitled “TILTED INTERFEROMETER,” by Henry A. Hill, the contents ofwhich is herein incorporated in their entirety by reference.

Other techniques for cyclic error compensation include those describedin U.S. Pat. No. 6,137,574 entitled “SYSTEMS AND METHODS FORCHARACTERIZING AND CORRECTING CYCLIC ERRORS IN DISTANCE MEASURING ANDDISPERSION INTERFEROMETRY;” U.S. Patent No. U.S. Pat. No. 6,252,668 B1,entitled “SYSTEMS AND METHODS FOR QUANTIFYING NON-LINEARITIES ININTERFEROMETRY SYSTEMS;” and U.S. Pat. No. 6,246,481, entitled “SYSTEMSAND METHODS FOR QUANTIFYING NONLINEARITIES IN INTERFEROMETRY SYSTEMS,”wherein all three are by Henry A. Hill, the contents of the threeabove-cited patents and patent applications are herein incorporated intheir entirety by reference.

Improved statistical accuracy in measured values of SDP can be obtainedby taking advantage of the relatively low bandwidth of measured valuesof SDP compared to the bandwidth of the corresponding lineardisplacement measurements using averaging or low pass filtering. Therelatively low bandwidth arises because of SDP invariance with respectto displacements of the mirror along the measurement axes and itinvariance (at least to second order) on rotations of the mirror.Variations in SDP occur primarily as a result of variations of themirror surface figure as the mirror is scanned orthogonally to themeasurement axes, which depends on the mirror scan speed, but generallyoccurs at much lower rates than the sampling rates of the detectors usedin the three-axis/plane interferometer. For example, where the 1/e² beamdiameter is 5 mm and the stage is scanned at a rate of 0.5 m/s, thebandwidth of measured values of SDP will be of the order of 100 Hzcompared to typical sampling rates of the respective detectors of theorder of 10 MHz.

The effects of offset errors in the measured values of SDP can bemeasured by use of an artifact wafer and the artifact wafer rotated by180 degrees and the offset error effects compensated. Offset errorsarise in SDP because SDP is derived from three different interferometermeasurements where each of the three interferometers can only measurerelative changes in a respective reference and measurement beam paths.In addition, the offset errors may change with time because thecalibrations of each of the three interferometers may change withrespect to each other, e.g. due to changes in temperature. An artifactwafer is a wafer that includes several alignment marks that areprecisely spaced with respect to each other. Accordingly, a metrologysystem can be calibrated by locating the marks with the system'salignment sensor and comparing the displacement between each alignmentmark as measured using the metrology system with the known displacement.Because of offset errors in the interferometer distance measurements andbecause the angle between the measurement axes of the x-axis and y-axisinterferometers is not generally known, the angle between the x-axis andy-axis stage mirrors should be independently measured. This angle can bemeasured in one or more planes according to whether interferometersystem 10 comprises one or two three-axis/plane interferometers by useof an artifact wafer and the artifact wafer rotated by 90 degrees.

While the foregoing description is with regard to a particularinterferometer assembly, namely interferometer 100, in general, otherassemblies can also be used to obtain values for SDP and otherparameters. For example, in some embodiments, an interferometer assemblycan be configured to monitor the position of a measurement object alongmore than three coplanar axes (e.g., four or more axes, five or moreaxes). Moreover, while interferometer includes non-coplanar measurementaxes, other embodiments can include exclusively coplanar axes.Furthermore, the relative position of the common measurement beam pathis not limited to the position in interferometer 100. For example, insome embodiments, the common measurement beam path can be an outermostpath, instead of being flanked by beam paths on either side within thecommon plane.

In certain embodiments, individual, rather than compound, opticalcomponents can be used. For example, free-standing beam splitters can beused to divide the first path measurement beam into the othermeasurement beams. Such a configuration may allow one to adjust therelative spacing of the beams, and hence the relative spacing of themeasurement axes in a multi-axis interferometer.

In some embodiments, multiple single axis interferometers can be usedinstead of a multi-axis interferometer. For example, a three-axis/planeinterferometer can be replaced by three single axis interferometers(e.g., high stability plane mirror interferometers), arranged so thateach interferometer's axis lies in a common plane.

As discussed previously, lithography tools are especially useful inlithography applications used in fabricating large scale integratedcircuits such as computer chips and the like. Lithography is the keytechnology driver for the semiconductor manufacturing industry. Overlayimprovement is one of the five most difficult challenges down to andbelow 100 nm line widths (design rules), see, for example, theSemiconductor Industry Roadmap, p. 82 (1997).

Overlay depends directly on the performance, i.e., accuracy andprecision, of the distance measuring interferometers used to positionthe wafer and reticle (or mask) stages. Since a lithography tool mayproduce $50–100M/year of product, the economic value from improvedperformance distance measuring interferometers is substantial. Each 1%increase in yield of the lithography tool results in approximately$1M/year economic benefit to the integrated circuit manufacturer andsubstantial competitive advantage to the lithography tool vendor.

The function of a lithography tool is to direct spatially patternedradiation onto a photoresist-coated wafer. The process involvesdetermining which location of the wafer is to receive the radiation(alignment) and applying the radiation to the photoresist at thatlocation (exposure).

To properly position the wafer, the wafer includes alignment marks onthe wafer that can be measured by dedicated sensors. The measuredpositions of the alignment marks define the location of the wafer withinthe tool. This information, along with a specification of the desiredpatterning of the wafer surface, guides the alignment of the waferrelative to the spatially patterned radiation. Based on suchinformation, a translatable stage supporting the photoresist-coatedwafer moves the wafer such that the radiation will expose the correctlocation of the wafer.

During exposure, a radiation source illuminates a patterned reticle,which scatters the radiation to produce the spatially patternedradiation. The reticle is also referred to as a mask, and these termsare used interchangeably below. In the case of reduction lithography, areduction lens collects the scattered radiation and forms a reducedimage of the reticle pattern. Alternatively, in the case of proximityprinting, the scattered radiation propagates a small distance (typicallyon the order of microns) before contacting the wafer to produce a 1:1image of the reticle pattern. The radiation initiates photo-chemicalprocesses in the resist that convert the radiation pattern into a latentimage within the resist.

Interferometry metrology systems, such as those discussed previously,are important components of the positioning mechanisms that control theposition of the wafer and reticle, and register the reticle image on thewafer. If such interferometry systems include the features describedabove, the accuracy of distances measured by the systems can beincreased and/or maintained over longer periods without offlinemaintenance, resulting in higher throughput due to increased yields andless tool downtime.

In general, the lithography system, also referred to as an exposuresystem, typically includes an illumination system and a waferpositioning system. The illumination system includes a radiation sourcefor providing radiation such as ultraviolet, visible, x-ray, electron,or ion radiation, and a reticle or mask for imparting the pattern to theradiation, thereby generating the spatially patterned radiation. Inaddition, for the case of reduction lithography, the illumination systemcan include a lens assembly for imaging the spatially patternedradiation onto the wafer. The imaged radiation exposes resist coatedonto the wafer. The illumination system also includes a mask stage forsupporting the mask and a positioning system for adjusting the positionof the mask stage relative to the radiation directed through the mask.The wafer positioning system includes a wafer stage for supporting thewafer and a positioning system for adjusting the position of the waferstage relative to the imaged radiation. Fabrication of integratedcircuits can include multiple exposing steps. For a general reference onlithography, see, for example, J. R. Sheats and B. W. Smith, inMicrolithography: Science and Technology (Marcel Dekker, Inc., New York,1998), the contents of which is incorporated herein by reference.

Interferometry systems described above can be used to precisely measurethe positions of each of the wafer stage and mask stage relative toother components of the exposure system, such as the lens assembly,radiation source, or support structure. In such cases, theinterferometry system can be attached to a stationary structure and themeasurement object attached to a movable element such as one of the maskand wafer stages. Alternatively, the situation can be reversed, with theinterferometry system attached to a movable object and the measurementobject attached to a stationary object.

More generally, such interferometry systems can be used to measure theposition of any one component of the exposure system relative to anyother component of the exposure system, in which the interferometrysystem is attached to, or supported by, one of the components and themeasurement object is attached, or is supported by the other of thecomponents.

Another example of a lithography tool 1100 using an interferometrysystem 1126 is shown in FIG. 4. The interferometry system is used toprecisely measure the position of a wafer (not shown) within an exposuresystem. Here, stage 1122 is used to position and support the waferrelative to an exposure station. Scanner 1100 includes a frame 1102,which carries other support structures and various components carried onthose structures. An exposure base 1104 has mounted on top of it a lenshousing 1106 atop of which is mounted a reticle or mask stage 1116,which is used to support a reticle or mask. A positioning system forpositioning the mask relative to the exposure station is indicatedschematically by element 1117. Positioning system 1117 can include,e.g., piezoelectric transducer elements and corresponding controlelectronics. Although, it is not included in this described embodiment,one or more of the interferometry systems described above can also beused to precisely measure the position of the mask stage as well asother moveable elements whose position must be accurately monitored inprocesses for fabricating lithographic structures (see supra Sheats andSmith Microlithozraphy: Science and Technology).

Suspended below exposure base 1104 is a support base 1113 that carrieswafer stage 1122. Stage 1122 includes a plane mirror 1128 for reflectinga measurement beam 1154 directed to the stage by interferometry system1126. A positioning system for positioning stage 1122 relative tointerferometry system 1126 is indicated schematically by element 1119.Positioning system 1119 can include, e.g., piezoelectric transducerelements and corresponding control electronics. The measurement beamreflects back to the interferometry system, which is mounted on exposurebase 1104. The interferometry system can be any of the embodimentsdescribed previously.

During operation, a radiation beam 1110, e.g., an ultraviolet (UV) beamfrom a UV laser (not shown), passes through a beam shaping opticsassembly 1112 and travels downward after reflecting from mirror 1114.Thereafter, the radiation beam passes through a mask (not shown) carriedby mask stage 1116. The mask (not shown) is imaged onto a wafer (notshown) on wafer stage 1122 via a lens assembly 1108 carried in a lenshousing 1106. Base 1104 and the various components supported by it areisolated from environmental vibrations by a damping system depicted byspring 1120.

In other embodiments of the lithographic scanner, one or more of theinterferometry systems described previously can be used to measuredistance along multiple axes and angles associated for example with, butnot limited to, the wafer and reticle (or mask) stages. Also, ratherthan a UV laser beam, other beams can be used to expose the waferincluding, e.g., x-ray beams, electron beams, ion beams, and visibleoptical beams.

In some embodiments, the lithographic scanner can include what is knownin the art as a column reference. In such embodiments, theinterferometry system 1126 directs the reference beam (not shown) alongan external reference path that contacts a reference mirror (not shown)mounted on some structure that directs the radiation beam, e.g., lenshousing 1106. The reference mirror reflects the reference beam back tothe interferometry system. The interference signal produce byinterferometry system 1126 when combining measurement beam 1154reflected from stage 1122 and the reference beam reflected from areference mirror mounted on the lens housing 1106 indicates changes inthe position of the stage relative to the radiation beam. Furthermore,in other embodiments the interferometry system 1126 can be positioned tomeasure changes in the position of reticle (or mask) stage 1116 or othermovable components of the scanner system. Finally, the interferometrysystems can be used in a similar fashion with lithography systemsinvolving steppers, in addition to, or rather than, scanners.

As is well known in the art, lithography is a critical part ofmanufacturing methods for making semiconducting devices. For example,U.S. Pat. No. 5,483,343 outlines steps for such manufacturing methods.These steps are described below with reference to FIGS. 5 a and 5 b.FIG. 5 a is a flow chart of the sequence of manufacturing asemiconductor device such as a semiconductor chip (e.g., IC or LSI), aliquid crystal panel or a CCD. Step 1151 is a design process fordesigning the circuit of a semiconductor device. Step 1152 is a processfor manufacturing a mask on the basis of the circuit pattern design.Step 1153 is a process for manufacturing a wafer by using a materialsuch as silicon.

Step 1154 is a wafer process which is called a pre-process wherein, byusing the so prepared mask and wafer, circuits are formed on the waferthrough lithography. To form circuits on the wafer that correspond withsufficient spatial resolution those patterns on the mask,interferometric positioning of the lithography tool relative the waferis necessary. The interferometry methods and systems described hereincan be especially useful to improve the effectiveness of the lithographyused in the wafer process.

Step 1155 is an assembling step, which is called a post-process whereinthe wafer processed by step 1154 is formed into semiconductor chips.This step includes assembling (dicing and bonding) and packaging (chipsealing). Step 1156 is an inspection step wherein operability check,durability check and so on of the semiconductor devices produced by step1155 are carried out. With these processes, semiconductor devices arefinished and they are shipped (step 1157).

FIG. 5 b is a flow chart showing details of the wafer process. Step 1161is an oxidation process for oxidizing the surface of a wafer. Step 1162is a CVD process for forming an insulating film on the wafer surface.Step 1163 is an electrode forming process for forming electrodes on thewafer by vapor deposition. Step 1164 is an ion implanting process forimplanting ions to the wafer. Step 1165 is a resist process for applyinga resist (photosensitive material) to the wafer. Step 1166 is anexposure process for printing, by exposure (i.e., lithography), thecircuit pattern of the mask on the wafer through the exposure apparatusdescribed above. Once again, as described above, the use of theinterferometry systems and methods described herein improve the accuracyand resolution of such lithography steps.

Step 1167 is a developing process for developing the exposed wafer. Step1168 is an etching process for removing portions other than thedeveloped resist image. Step 1169 is a resist separation process forseparating the resist material remaining on the wafer after beingsubjected to the etching process. By repeating these processes, circuitpatterns are formed and superimposed on the wafer.

The interferometry systems described above can also be used in otherapplications in which the relative position of an object needs to bemeasured precisely. For example, in applications in which a write beamsuch as a laser, x-ray, ion, or electron beam, marks a pattern onto asubstrate as either the substrate or beam moves, the interferometrysystems can be used to measure the relative movement between thesubstrate and write beam.

As an example, a schematic of a beam writing system 1200 is shown inFIG. 6. A source 1210 generates a write beam 1212, and a beam focusingassembly 1214 directs the radiation beam to a substrate 1216 supportedby a movable stage 1218. To determine the relative position of thestage, an interferometry system 1220 directs a reference beam 1222 to amirror 1224 mounted on beam focusing assembly 1214 and a measurementbeam 1226 to a mirror 1228 mounted on stage 1218. Since the referencebeam contacts a mirror mounted on the beam focusing assembly, the beamwriting system is an example of a system that uses a column reference.Interferometry system 1220 can be any of the interferometry systemsdescribed previously. Changes in the position measured by theinterferometry system correspond to changes in the relative position ofwrite beam 1212 on substrate 1216. Interferometry system 1220 sends ameasurement signal 1232 to controller 1230 that is indicative of therelative position of write beam 1212 on substrate 1216. Controller 1230sends an output signal 1234 to a base 1236 that supports and positionsstage 1218. In addition, controller 1230 sends a signal 1238 to source1210 to vary the intensity of, or block, write beam 1212 so that thewrite beam contacts the substrate with an intensity sufficient to causephotophysical or photochemical change only at selected positions of thesubstrate.

Furthermore, in some embodiments, controller 1230 can cause beamfocusing assembly 1214 to scan the write beam over a region of thesubstrate, e.g., using signal 1244. As a result, controller 1230 directsthe other components of the system to pattern the substrate. Thepatterning is typically based on an electronic design pattern stored inthe controller. In some applications the write beam patterns a resistcoated on the substrate and in other applications the write beamdirectly patterns, e.g., etches, the substrate.

An important application of such a system is the fabrication of masksand reticles used in the lithography methods described previously. Forexample, to fabricate a lithography mask an electron beam can be used topattern a chromium-coated glass substrate. In such cases where the writebeam is an electron beam, the beam writing system encloses the electronbeam path in a vacuum. Also, in cases where the write beam is, e.g., anelectron or ion beam, the beam focusing assembly includes electric fieldgenerators such as quadrapole lenses for focusing and directing thecharged particles onto the substrate under vacuum. In other cases wherethe write beam is a radiation beam, e.g., x-ray, UV, or visibleradiation, the beam focusing assembly includes corresponding optics andfor focusing and directing the radiation to the substrate.

A number of embodiments of the invention have been described.Nevertheless, it will be understood that various modifications may bemade without departing from the spirit and scope of the invention.Accordingly, other embodiments are within the scope of the followingclaims.

1. A method, comprising: interferometrically monitoring a distancebetween an interferometry assembly and a measurement object along eachof three different measurement axes while moving the measurement objectrelative to the interferometry assembly; determining values of aparameter for different positions of the measurement object from themonitored distances, wherein for a given position the parameter is basedon the distances of the measurement object along each of the threedifferent measurement axes at the given position; and derivinginformation about a surface figure profile of the measurement objectfrom a frequency transform of at least the parameter values.
 2. Themethod of claim 1, wherein the frequency transform is a Fouriertransform.
 3. The method of claim 2, wherein the Fourier transform is adiscrete Fourier transform.
 4. The method of claim 2, derivinginformation about the surface figure profile of the measurement objectcomprises determining Fourier coefficients for a function characterizingthe surface figure profile.
 5. The method of claim 1, wherein theinformation about the surface figure profile is derived from a frequencytransform of values of a second parameter, wherein for a given positionthe second parameter is based on the monitored distances of themeasurement object along each of two of the measurement axes at thegiven position.
 6. The method of claim 5, wherein the second parameteris also based on the monitored distance of the measurement object alongone of the measurement axes at another position.
 7. The method of claim6, wherein determining the value of the second parameter for eachposition of the measurement object includes calculating a differencebetween the monitored distance along two of the measurement axes formultiple different positions of the measurement object.
 8. The method ofclaim 1, wherein monitoring the distance between the interferometryassembly and the measurement object comprises simultaneously measuring alocation of the measurement object along each of the measurement axesfor each of the different positions of the measurement object.
 9. Themethod of claim 1, further comprising determining values of a thirdparameter for different positions of the measurement object, wherein foreach position the third parameter is based on the distance of themeasurement object along each of two of the measurement axes at thatposition and the distance of the measurement object along one of themeasurement axes at another position.
 10. The method of claim 1, furthercomprising using the information about the surface figure profile of themeasurement object to improve the accuracy of measurements made usingthe interferometry assembly.
 11. The method of claim 1, furthercomprising using a lithography tool to expose a substrate with radiationpassing through a mask while interferometrically monitoring the distancebetween the interferometry assembly and the measurement object, whereinthe position of the substrate or the mask relative to a reference frameis related to the distance between the interferometry assembly and themeasurement object.
 12. The method of claim 1, wherein theinterferometer assembly or the measurement object are attached to astage and at least one of the monitored distances is used to monitor theposition of the stage relative to a frame supporting the stage.
 13. Alithography method for use in fabricating integrated circuits on awafer, the method comprising: supporting the wafer on a moveable stage;imaging spatially patterned radiation onto the wafer; adjusting theposition of the stage; and monitoring the position of the stage usingthe measurement object and using the information about the surfacefigure profile of the measurement object derived using the method ofclaim 1 to improve the accuracy of the monitored position of the stage.14. A lithography method for use in the fabrication of integratedcircuits comprising: directing input radiation through a mask to producespatially patterned radiation; positioning the mask relative to theinput radiation; monitoring the position of the mask relative to theinput radiation using the measurement object and using the informationabout the surface figure profile of the measurement object derived usingthe method of claim 1 to improve the accuracy of the monitored positionof the mask; and imaging the spatially patterned radiation onto a wafer.15. A lithography method for fabricating integrated circuits on a wafercomprising: positioning a first component of a lithography systemrelative to a second component of a lithography system to expose thewafer to spatially patterned radiation; and monitoring the position ofthe first component relative to the second component using themeasurement object and using the information about the surface figureprofile of the measurement object derived using the method of claim 1 toimprove the accuracy of the monitored position of the first component.16. A method for fabricating integrated circuits, the method comprisingthe lithography method of claim
 13. 17. A method for fabricatingintegrated circuits, the method comprising the lithography method ofclaim
 14. 18. A method for fabricating integrated circuits, the methodcomprising the lithography method of claim
 15. 19. A method forfabricating a lithography mask, the method comprising: directing a writebeam to a substrate to pattern the substrate; positioning the substraterelative to the write beam; and monitoring the position of the substraterelative to the write beam using the measurement object and using theinformation about the surface figure profile of the measurement objectderived using the method of claim 1 to improve the accuracy of themonitored position of the substrate.
 20. The method of claim 1,furthercomprising moving the measurement object relative to the interferometryassembly in response to the output signal.
 21. The method of claim 1,wherein the output signal is used to provide the information about thesurface figure profile to an electronic controller's memory or to disk.22. An apparatus, comprising: a interferometer assembly configured toproduce three output beams each including interferometric informationabout a distance between the interferometer and a measurement objectalong a respective axis; and an electronic processor configured todetermine values of a parameter or different positions of themeasurement object from the interferometric information, wherein for agiven position the parameter is based on the distances of themeasurement object along each of the respective measurement axes at thegiven position, the electronic processor being further configured toderive information about a surface figure profile of the measurementobject from a frequency transform of at least the parameter values. 23.The apparatus of claim 22, wherein the measurement object is a planemirror measurement object.
 24. The apparatus of claim 22, wherein thethree output beams each include a component that makes one pass to themeasurement object along a common beam path.
 25. The apparatus of claim22, wherein the measurement axes are co-planar.
 26. The apparatus ofclaim 22, wherein the measurement axes are parallel.
 27. The apparatusof claim 22, further comprising a stage and wherein the interferometerassembly or measurement object are attached to the stage.
 28. Alithography system for use in fabricating integrated circuits on awafer, the system comprising: a stage for supporting the wafer; anillumination system for imaging spatially patterned radiation onto thewafer; a positioning system for adjusting the position of the stagerelative to the imaged radiation; and the apparatus of claim 22 formonitoring the position of the wafer relative to the imaged radiation.29. A lithography system for use in fabricating integrated circuits on awafer, the system comprising: a stage for supporting the wafer; and anillumination system including a radiation source, a mask, a positioningsystem, a lens assembly, and the apparatus of claim 22, wherein duringoperation the source directs radiation through the mask to producespatially patterned radiation, the positioning system adjusts theposition of the mask relative to the radiation from the source, the lensassembly images the spatially patterned radiation onto the wafer, andthe apparatus monitors the position of the mask relative to theradiation from the source.
 30. A beam writing system for use infabricating a lithography mask, the system comprising: a sourceproviding a write beam to pattern a substrate; a stage supporting thesubstrate; a beam directing assembly for delivering the write beam tothe substrate; a positioning system for positioning the stage and beamdirecting assembly relative one another; and the apparatus of claim 22for monitoring the position of the stage relative to the beam directingassembly.
 31. A method for fabricating integrated circuits, the methodcomprising: applying a resist to a wafer; forming a pattern of a mask inthe resist by exposing the wafer to radiation using the lithographysystem of claim 28; and producing an integrated circuit from the wafer.32. A method for fabricating integrated circuits, the method comprising:applying a resist to a wafer; forming a pattern of a mask in the resistby exposing the wafer to radiation using the lithography system of claim29; and producing an integrated circuit from the wafer.